Active beam shaping system and method using sequential deformable mirrors

ABSTRACT

An active optical beam shaping system includes a first deformable mirror arranged to at least partially intercept an entrance beam of light and to provide a first reflected beam of light, a second deformable mirror arranged to at least partially intercept the first reflected beam of light from the first deformable mirror and to provide a second reflected beam of light, and a signal processing and control system configured to communicate with the first and second deformable mirrors. The first deformable mirror, the second deformable mirror and the signal processing and control system together provide a large amplitude light modulation range to provide an actively shaped optical beam.

FEDERAL FUNDING BY THE U.S. GOVERNMENT

This invention was made with Government support of Grant No. NAS7-03001,awarded by the National Aeronautics and Space Administration (NASA). TheU.S. Government has certain rights in this invention.

BACKGROUND

1. Field of Invention

The field of the currently claimed embodiments of this invention relatesto active optical beam shaping systems and methods, and moreparticularly to active optical beam shaping systems and methods usingsequential deformable mirrors.

2. Discussion of Related Art

Exo-planetary systems that are directly imaged using existing facilities(Marois et al. 2008; Kalas et al. 2008; Lagrange et al. 2010) give aunique laboratory to constrain planetary formation at wide separations(Rafikov 2005; Dodson-Robinson et al. 2009; Kratter et al. 2010; Johnsonet al. 2010), to study the planetary luminosity distribution at criticalyoung ages (Spiegel & Burrows 2012; Fortney et al. 2008) and theatmospheric properties of low surface gravity objects (Barman et al.2011b,a; Marley et al. 2010; Madhusudhan et al. 2011). Upcoming surveys,conducted with instruments specifically designed for high-contrast(Dohlen et al. 2006; Graham et al. 2007; Hinkley et al. 2011), willunravel the bulk of this population of self-luminous Jovian planets andprovide an unprecedented understanding of their formation history. Suchinstruments will reach the contrast required to achieve their scientificgoals by combining Extreme Adaptive Optics systems (Ex-AO, Poyneer &V'eran (2005)), optimized coronagraphs (Soummer et al. 2011; Guyon 2003;Rouan et al. 2000) and nanometer class wavefront calibration (Sauvage etal. 2007; Wallace et al. 2009; Pueyo et al. 2010). In the future,high-contrast instruments on Extremely Large Telescopes will focus onprobing planetary formation in distant star forming regions (Macintoshet al. 2006), characterizing both the spectra of cooler gas giants(V'erinaud et al. 2010) and the reflected light of planets in thehabitable zone of low mass stars. The formidable contrast necessary toinvestigate the presence of biomarkers at the surface of earth analogs(>10¹⁰) cannot be achieved from the ground beneath atmosphericturbulence and will require dedicated space-based instruments (Guyon2005).

The coronagraphs that will equip upcoming Ex-AO instruments on 8 meterclass telescopes have been designed for contrasts of at most ˜10⁻⁷.Secondary support structure (or spiders: 4 struts each 1 cm wide, ˜0.3%of the total pupil diameter in the case of Gemini South) have a smallimpact on starlight extinction at such levels of contrasts. In thiscase, coronagraphs have thus been optimized on circularly symmetricapertures, which only take into account the central obscuration (Soummeret al. 2011). However, high-contrast instrumentation on futureobservatories will not benefit from such gentle circumstances. ELTs willhave to support a substantially heavier secondary than 8 meter classobservatories do, and over larger lengths: as a consequence, therelative area covered by the secondary support will increase by a factorof 10 (30 cm wide spiders, occupying ˜3% of the pupil diameter in thecase of TMT). This will degrade the contrast of coronagraphs onlydesigned for circularly obscured geometries by a factor ˜100, when theactual envisioned contrast for an ELT exo-planet imager can be as low as˜10⁻⁸ (Macintosh et al. 2006). While the trade-offs associated withminimization of spider width in the space-based case have yet to beexplored, secondary support structures will certainly hamper thecontrast depth of coronagraphic instruments of such observatories atlevels that are well above the 10¹⁰ contrast requirement. As aconsequence, telescope architectures currently envisioned for directcharacterization of exo-earths consist of monolithic, off-axis, and thusun-obscured, telescopes (Guyon et al. 2008; Trauger et al. 2010).Coronagraphs for such architectures take advantage of the pupil symmetryto reach a theoretical contrast of ten orders of magnitude (Guyon et al.2005; Vanderbei et al. 2003a,b; Kasdin et al. 2005; Mawet et al, 2010;Kuchner & Traub 2002; Soummer et al. 2003). However, using obscuredon-axis and/or segmented apertures take full advantage of the limitedreal estate associated with a given launch vehicle and can allow largerapertures that increase the scientific return of space-based directimaging survey. Recent solutions can mitigate the presence of secondarysupport structures in on-axis apertures. However these concepts presentpractical limitations: APLCs on arbitrary apertures (Soummer et al.2009) and Shaped Pupils (Carlotti et al. 2011) suffer from throughputloss for very high contrast designs, and PIAAMCM (Guyon et al. 2010a)rely on a phase mask technology whose chromatic properties have not yetbeen fully characterized. Moreover, segmentation will further complicatethe structure of the telescope's pupil: both the amplitudediscontinuities created by the segments gaps and the phasediscontinuities resulting from imperfect phasing will thus furtherdegrade coronagraphic contrast. Devising a practical solution forbroadband coronagraphy on asymmetric, unfriendly apertures is anoutstanding problem in high contrast instrumentation. Therefore, thereremains a need for improved beam shaping systems and methods.

SUMMARY

An active optical beam shaping system according to an embodiment of thecurrent invention includes a first deformable mirror arranged to atleast partially intercept an entrance beam of light and to provide afirst reflected beam of light, a second deformable mirror arranged to atleast partially intercept the first reflected beam of light from thefirst deformable mirror and to provide a second reflected beam of light,and a signal processing and control system configured to communicatewith the first and second deformable mirrors. The signal processing andcontrol system is configured to provide control signals to the firstdeformable mirror so as to configure a reflecting surface of the firstdeformable mirror to substantially conform to a calculated surface shapeso that the first reflected beam of light will have a predeterminedamplitude profile. The signal processing and control system is furtherconfigured to provide control signals to the second deformable mirror toconfigure a reflecting surface of the second deformable mirror tosubstantially conform to a calculated surface shape so that a phasedelay profile across the second reflected beam of light will besubstantially constant when a phase delay profile across the entrancebeam of light is substantially constant. The first deformable mirror,the second deformable mirror and the signal processing and controlsystem together provide a large amplitude light modulation range toprovide an actively shaped optical beam.

An optical transmitter according to an embodiment of the currentinvention includes an optical source, and an optical modulator arrangedin an optical path of the optical source. The optical modulator includesa first deformable mirror arranged to at least partially intercept anentrance beam of light from the optical source and to provide a firstreflected beam of light, a second deformable mirror arranged to at leastpartially intercept the first reflected beam of light from the firstdeformable mirror and to provide a second reflected beam of light, and asignal processing and control system configured to communicate with thefirst and second deformable mirrors. The signal processing and controlsystem is configured to provide control signals to the first deformablemirror and the second deformable mirror to provide at least a portion ofthe second reflected beam of light as an output beam having at least oneof a selectable amplitude profile or selectable phase profile.

An optical receiver according to an embodiment of the current inventionincludes an optical detector, and an optical filter arranged in anoptical path of the optical detector. The optical filter includes afirst deformable mirror arranged to at least partially intercept anentrance beam of light being detected to provide a first reflected beamof light, a second deformable mirror arranged to at least partiallyintercept the first reflected beam of light from the first deformablemirror and to provide a second reflected beam of light, and a signalprocessing and control system configured to communicate with the firstand second deformable mirrors. The signal processing and control systemis configured to provide control signals to the first deformable mirrorand the second deformable mirror to provide at least a portion of thesecond reflected beam of light as an output beam directed to the opticaldetector having at least one of a selectable amplitude profile orselectable phase profile.

An optical communication system according to an embodiment of thecurrent invention includes an optical transmitter, an opticaltransmission path optically coupled to the optical transmitter, and anoptical receiver optically coupled to the optical transmission path. Atleast one of the optical transmitter, the optical transmission path, orthe optical receiver includes an active optical beam shaper. The activeoptical beam shaper includes a first deformable mirror arranged to atleast partially intercept an entrance beam of light to provide a firstreflected beam of light, a second deformable mirror arranged to at leastpartially intercept the first reflected beam of light from the firstdeformable mirror and to provide a second reflected beam of light, and asignal processing and control system configured to communicate with thefirst and second deformable mirrors. The signal processing and controlsystem is configured to provide control signals to the first deformablemirror and the second deformable mirror to provide at least a portion ofthe second reflected beam of light as an output beam having at least oneof a selectable amplitude profile or selectable phase profile.

An optical telescope according to an embodiment of the current inventionincludes a light collection and magnification system, an active opticalbeam shaper arranged in an optical path of light collected and magnifiedby the light collection and magnification system, and an opticaldetection system arranged to receive a beam of light from the activeoptical beam shaper. The active optical beam shaper includes a firstdeformable mirror arranged to at least partially intercept an entrancebeam of light to provide a first reflected beam of light, a seconddeformable mirror arranged to at least partially intercept the firstreflected beam of light from the first deformable mirror and to providea second reflected beam of light, and a signal processing and controlsystem configured to communicate with the first and second deformablemirrors. The signal processing and control system is configured toprovide control signals to the first deformable mirror and the seconddeformable mirror to provide at least a portion of the second reflectedbeam of light as an output beam having at least one of a selectableamplitude profile or selectable phase profile.

A solar furnace according to an embodiment of the current inventionincludes a light collection system, an active optical beam shaperarranged in an optical path of light collected by the light collectionsystem, and an optical focusing system arranged to receive a beam oflight from the active optical beam shaper to be focused on an object tobe heated. The active optical beam shaper includes a first deformablemirror arranged to at least partially intercept an entrance beam oflight to provide a first reflected beam of light, a second deformablemirror arranged to at least partially intercept the first reflected beamof light from the first deformable mirror and to provide a secondreflected beam of light, and a signal processing and control systemconfigured to communicate with the first and second deformable mirrors.The signal processing and control system is configured to providecontrol signals to the first deformable mirror and the second deformablemirror to provide at least a portion of the second reflected beam oflight as an output beam having at least one of a selectable amplitudeprofile or selectable phase profile.

An optical pulse-shaping system according to an embodiment of thecurrent invention includes an optical pulse source, a first diffractiongrating disposed in an optical path of the optical pulse source, anactive optical beam shaper disposed in an optical path of lightdiffracted from the first diffraction grating, and a second diffractiongrating disposed in an optical path of light output from the activeoptical beam shaper. The active optical beam shaper includes a firstdeformable mirror arranged to at least partially intercept an entrancebeam of light to provide a first reflected beam of light, a seconddeformable mirror arranged to at least partially intercept the firstreflected beam of light from the first deformable mirror and to providea second reflected beam of light, and a signal processing and controlsystem configured to communicate with the first and second deformablemirrors. The signal processing and control system is configured toprovide control signals to the first deformable mirror and the seconddeformable mirror to provide at least a portion of the second reflectedbeam of light as an output beam having at least one of a selectableamplitude profile or selectable phase profile.

BRIEF DESCRIPTION OF THE DRAWINGS

Further objectives and advantages will become apparent from aconsideration of the description, drawings, and examples.

FIG. 1A is a schematic illustration of an active optical beam shapingsystem according to an embodiment of the current invention.

FIG. 1B is a schematic illustration of an optical transmitter accordingto an embodiment of the current invention.

FIG. 1C is a schematic illustration of laser beam launch systemaccording to an embodiment of the current invention.

FIG. 1D is a schematic illustration of an optical receiver according toan embodiment of the current invention.

FIG. 1E is a schematic illustration of a solar furnace according to anembodiment of the current invention.

FIG. 1F is a schematic of a beam shaping device according to anembodiment of the current invention. Some advantages of beam shaping caninclude: (1) Can put more energy in a pulse. These can be ultra-short,high energy femtosecond pulses. Can include applications to laserfusion. (2) Can actively correct for beam aberrations that do notdegrade the gratings,

FIG. 1G is a schematic illustration of a telescope according to anembodiment of the current invention. Top: Architecture applicable tofuture exo-earth imaging missions: a monolithic un-obscured telescopefeeds a coronagraph designed on a circular aperture. The wavefronterrors are corrected using two sequential Deformable Mirrors (DMs) thatare controlled using a quasi-linear feedback loop based on image planediagnostics. The propagation between optical surfaces, between the DMsin particular is assumed to occur in the Fresnel regime. Bottom: ACADsolution: the coronagraph is designed for a circular geometry around thecentral obscuration. The two sequential DMs are controlled in thenon-linear regime based on a pupil plane cost function. The propagationbetween the DMs is modeled using ray optics, and we conduct aquantitative one-dimensional analysis of the diffraction artifacts.

FIG. 1H is a schematic illustration to help explain some concepts ofsome embodiments of the current invention.

FIG. 2 shows optimally designed Apodized Pupil Lyot Coronagraphs oncircularly obscured apertures. With fixed obscuration ratio, size ofopaque focal plane mask, IWA and OWA, our linear programming approachyields solutions with theoretical contrast below 10¹⁰. All PSFs shown onthis figure are monochromatic for λ=λ₀. When all other quantities remainequal and the central obscuration ratio increases (from 10% in the toppanel to 20% in the middle panel), then the solution becomes moreoscillatory (e.g less feasible) and the contrast constraint has to berelaxed. Eventually the optimizer does not find a solution and the sizeof the opaque focal plane mask (and thus the IWA) has to be increased(central obscuration of 30% in the bottom panel). On the right handside, we present our results in two configurations: when the apodizationis achieved using at grayscale screen (APLC), “on-sky” λ/D bottomx-axis, and when the apodization is achieved via two pupil remappingmirrors (PIAAC), “on-sky” λ/D top x-axis. We adopt this presentation toshow that ACAD is “coronagraph independent” and that it can be appliedto coronagraphs with high throughput and small IWA.

FIG. 3 shows a schematic of the optical system considered: the telescopeaperture is followed by two sequential Deformable Mirrors (DMs) innon-conjugate planes whose purpose is to remap the pupildiscontinuities. The beam then enters a coronagraph to suppress the bulkof the starlight: in this figure we show an Apodized Lyot PupilCoronagraph (APLC). This is the coronagraphic architecture we considerfor the remainder of the paper but we stress that the method presentedherein is applicable to any coronagraph.

FIG. 4 Top: In the ideal case, the two DMs would fully remap all thediscontinuities in the telescope's aperture to feed a fully uniform beamto the coronagraph. However this would require discontinuities in themirror's curvatures which are not achieved in practice. Moreover solvingthe Monge-Ampere Equation in this direction is a difficult exercise asthe right hand side of Eq. 26 presents an implicit dependence on thesolution h₂.

Bottom: We circumvent this problem by solving the reverse problem, wherethe the input beam is now uniform and the implicit dependence drops out.Moreover we taper the edges of the discontinuities by convolving thetarget field A(x₁,y₁) by a gaussian of full width at half maximum ω(ω=50 cycles per aperture in this figure).

FIG. 5 shows forward and reverse coordinate systems, with theirrespective forward and reverse coordinate transforms, in the case of thefolded system studied in this example. Note that this figure showsnormalized units. As explained in the body of the text thecorrespondence between normalized and real units scales as follow:(x_(i),y_(i))=(DX_(i),DY_(i)), (f_(i),g_(i))=(DF_(i),DG_(i)),

${H_{i} = {\frac{D^{2}}{z}h_{i}}},$

where i=1, 2, D is the aperture diameter and Z the separation betweenDMs. We solve the Monge-Ampere Equation in normalized coordinates andthen apply the scalings in order to find the true DM shapes.

FIG. 6 shows the virtual field at M1 when solving the reverse problem,A(X₁,Y₁) is the desired apodization (top), A_(n)(X₁,Y₁) is theapodization obtained after solving the Monge-Ampere Equation (center).The bottom panel shows the difference between the two quantities: thebulk of the energy in the residual is located in high spatialfrequencies that cannot be controlled by the DMs.

FIG. 7 shows boundary conditions seen in the horizontal remapped space.Should the boundary conditions have strictly been enforced by our solverthen

${{F_{2}\left( {{- \frac{1}{2}},Y} \right)} = {- \frac{1}{2}}},{{F_{2}\left( {\frac{1}{2},Y} \right)} = \frac{1}{2}},{{F_{2}\left( {X,{- \frac{1}{2}}} \right)} = X},{{F_{2}\left( {X,\frac{1}{2}} \right)} = {X.}}$

The remapping function obtained with our solutions is very close tothese theoretical boundary conditions and the residuals can easily bemitigated by sacrificing the edge rows and columns of actuators on eachDM.

FIG. 8 shows boundary conditions seen in the vertical remapped space.Should the boundary conditions have strictly been enforced by our solverthen

${{G_{2}\left( {{- \frac{1}{2}},Y} \right)} = Y},{{G_{2}\left( {\frac{1}{2},Y} \right)} = Y},{{G_{2}\left( {X,{- \frac{1}{2}}} \right)} = {- \frac{1}{2}}},{{G_{2}\left( {X,\frac{1}{2}} \right)} = {\frac{1}{2}.}}$

The remapping function obtained with our solutions is very close tothese theoretical boundary conditions and the residuals can easily bemitigated by sacrificing the edge rows and columns of actuators on eachDM.

FIG. 9 shows beam amplitude before and after the DMs in the case of ageometry similar to JWST. We chose to solve the reverse problem over asquare using a Fourier basis set and not strictly enforcing boundaryconditions. This results in small distortions of the edges of the actualaperture in the vicinity of segments gaps and secondary supports. Weaddress this problem by slightly oversizing the secondary obscurationand undersizing the aperture edge in the coronagraph.

FIG. 10, Left: Comparison of edge diffraction between PIAA and Fresnelpropagation. Because the discontinuities in the pupil occur at thelocation where Γ_(x)<1, a PIAA amplifies the chromatic ringing whencompared to a more classical propagation. Right: Comparison of edgediffraction between ACAD and Fresnel propagation. Because thediscontinuities in the pupil occur at the location where Γ_(x)>1, a ACADdamps the chromatic ringing when compared to a more classicalpropagation. These simulations were carried out for a one dimensionalaperture. In this case the Monge-Ampere Equation can be solved usingfinite elements and the calculation of the diffractive effects reducesto the evaluation of various Fresnel special functions at varyingwavelength. The full two-dimensional problem is not separable andrequires the development of novel numerical tools.

FIG. 11 Results obtained when applying our approach to a geometrysimilar to JWST We used two 3 cm DMs of 64 actuators separated by 1 m.Their maximal surface deformation is 1.1 μm, well within the strokelimit of current DM technologies. The residual light in the correctedPSF follows the secondary support structures and can potentially befurther cancelled by controlling the DMs using an image plane based costfunction, see FIG. 23.

FIG. 12 Case of JWST: Radial average obtained when applying ACAD. Weused two 3 cm DMs of 64 actuators separated by 1 m. Their maximalsurface deformation is 1.1 μm, well within the stroke limit of currentDM technologies. ACAD yields a gain in contrast of two orders ofmagnitude, and provides contrasts levels similar to upcoming Ex-AOinstruments, which are designed on much friendlier apertures geometries.Since ACAD removes the bulk of the light diffracted by the asymmetricaperture discontinuities, the final contrast can be improved bycontrolling the DMs using and image plane based cost function, see FIG.23.

FIG. 13 shows results obtained when applying our approach to a TMTgeometry. We used two 3 cm DMs of 64 actuators separated by 1 m. Theirmaximal surface deformation is 0.9μ 0.9 μm, well within the stroke limitof current DM technologies. The final contrast is below 10⁷ 10⁷, in aregime favorable for direct imaging of exo-planets with ELTs.

FIG. 14 Case of TMT: radial average obtained when applying ACAD. We usedtwo 3 cm DMs of 64 actuators separated by 1 m. Their maximal surfacedeformation is 0.9 μ 0.9 μm, well within the stroke limit of current DMtechnologies. The final contrast is below 1.0⁷ 10⁷, in a regimefavorable for direct imaging of exo-planets with ELTs. Since ACADremoves the bulk of the light diffracted by the asymmetric aperturediscontinuities, the final contrast can be enhanced by controlling theDMs using and image plane based metric.

FIG. 15 shows PSFs resulting from ACAD when varying the number andthickness of secondary support structures while maintaining theircovered surface constant. The surface area covered in this example is50% greater than in the TMT example shown on FIG. 13. As the spiders getthinner their impact on raw contrast becomes smaller and the starlightsuppression after DM correction becomes bigger. For a relatively smallnumber of spiders (<12) the contrast improvement on each singlestructure is the dominant phenomenon, regardless of the number ofspiders. ELTs designed with a moderate to large number of thin secondarysupport structure (6 to 12) present aperture discontinuities which areeasy to correct with ACAD.

FIG. 16 shows radial PSF profiles resulting from ACAD when varying thenumber and thickness of secondary support structures while maintainingtheir covered surface constant. The surface area covered in this exampleis 50% greater than in the TMT example shown on FIG. 13. As the spidersget thinner their impact on raw contrast becomes lesser and thestarlight suppression after DM correction becomes greater. In the 12spiders example, at large separations, the average contrast is an orderof magnitude higher than reported on FIG. 14.

FIG. 17 shows PSFs resulting from ACAD when varying the number andthickness of secondary support structures while maintaining theircovered surface constant. The surface area covered in this example is50% greater than in the TMT example shown on FIG. 13. When the number ofspiders increases, they produce a sharp circular diffraction feature atN_(Spiders)/π λ/D. If this number is greater than the size of the focalplane mask this structure appears in the high contrast zone and is verydifficult to correct with ACAD. The brightness of this structure ismitigated by the fact that the spiders are very thin.

FIG. 18 shows radial PSF profiles resulting from ACAD when varying thenumber and thickness of secondary support structures while maintainingtheir covered surface constant. The surface area covered in this exampleis 50% greater than in the TMT example shown on FIG. 13. With a largenumber of spiders the bright ring in the PSF structure located atN_(Spiders)/π λ/D is difficult to correct with ACAD. However, since thespiders become thinner their net effect on contrast after ACAD remainssmall.

FIG. 19 shows PSFs resulting from ACAD when varying the number andthickness of secondary support structures. As the spiders get thinnertheir impact on raw contrast becomes lesser and the starlightsuppression after DM correction becomes greater. In this case ω wasoptimized on a very fine grid and the aperture we clocked in a favorabledirection with respect to the Fourier basis.

FIG. 20 shows PFSs resulting from ACAD when varying the number andthickness of secondary support structures. As the spiders get thinnertheir impact on raw contrast becomes lesser and the starlightsuppression after DM correction becomes greater. In this case ω wasoptimized on a very fine grid and the aperture we clocked in a favorabledirection with respect to the Fourier basis. Even for spiders as thickas 0.5% of the telescope aperture the designed contrast of thecoronagraph is retrieved.

FIG. 21 shows off-axis PSF after ACAD in the case of a geometry similarto JWST. The aspheric surface of the DMs introduces a slightfield-dependent distortion. However the core of the PSF is stillconcentrated within the central airy disk and the DMs only have aneffect on the PSF tail. Field distortion does not thus hamper thedetectability of faint off-axis sources.

FIG. 22 shows broadband wavefront correction (20% bandwidth around 700nm) with a single DM in segmented telescope with discontinuous surfaceerrors. Top Left: wavefront before the coronagraph. Top Right: broadbandaberrated PSF with DM at rest. Bottom Left: DM surface resulting fromthe wavefront control algorithm. Bottom Right: broadband corrected PSF.Note that the wavefront control algorithm seeks to compensate for thediffractive artifacts associated with the secondary support structures:it attenuates them on the right side of the PSF while it strengthensthem on the left side of the PSF. As a result the DM surface becomes toolarge at the pupil spider's location and the quasi-linear wavefrontcontrol algorithm eventually diverges.

FIG. 23 shows broadband wavefront correction (20% bandwidth around 700nm) in a segmented telescope whose pupil has been re-arranged usingACAD. The surface of the first DM is set according to the ACADequations. The surface of the second DM is the sum of the ACAD solutionand a small perturbation calculated using a quasi-linear wavefrontcontrol algorithm. Top Left: wavefront before the coronagraph. Note thatthe ACAD remapping has compressed the wavefront errors near the strutsand the segment gaps. Top Right: broadband aberrated PSF with DMs set tothe ACAD solution. Bottom Left: perturbation of DM2's surface resultingfrom the wavefront control algorithm. Bottom Right: broadband correctedPSF. The wavefront control algorithm now yields a DM surface that doesnot feature prominent deformations at the location of the spiders. Mostof the DM stroke is located at the edge of the segments, at location ofthe wavefront discontinuities. There, the DM surface eventually becomestoo large and the quasi-linear wavefront control algorithm diverges.However this results in higher contrasts than in the absence of ACAD.

FIG. 24 shows radial average in the half dark plane of the PSFs on FIG.22 and FIG. 23. In the presence of wavefront discontinuities correctedusing a continuous membrane DM, ACAD still yields, over a 20% bandwidtharound 700 nm, PSF with a contrast 100 larger than in a classicalsegmented telescope. Moreover this figure illustrates that since it isbased on a true image plane metric, the wavefront control algorithm canbe used (within the limits of its linear regime) to improve upon theACAD DM shapes derived solving the Monge-Ampere Equation.

FIG. 25 shows further embodiments for higher contrasts with ACAD. Theblue and orange colors respectively represent the current state of theart in wavefront control and the work described in the presentmanuscript, as in FIG. 1G. In brown are listed avenues to further thecontrasts presented herein: 1) combining ACAD with coronagraphs designedon segmented and/or on-axis apertures, 2) using diffractive models toclose a quasi-linear focal plane based loop using a metric whosestarting point corresponds to the DM shapes calculated in the non-linearregime.

FIG. 26 shows simulation of ACAD in the SR-Fresnel approximation. Wesimulated the propagation between DM1 and DM2 according to Eq. (59), andsimulated the rest of the coronagraphic train using the Fresnelapproximation. Fresnel propagation simply replicates the input pupil's(top panels) deep and sharp discontinuities in the final reimaged pupil(right bottom panel). ACAD SR-Fresnel instead transforms them intoshallow and smooth mid-spatial frequency ripples (left bottom panel).The chromatic nature of these oscillations sets the ultimate bandwidthof ACAD, but unlike in PIAA coronagraphs the beam reshaping here doesnot amplify these oscillations any beyond normal edge ringing for out ofpupil optics in the Fresnel regime (discussed in Pueyo and Kasdin,2007). The plotted slices across the secondary support structures(middle panels) show that ripples in the SR-Fresnel approximation areneither bigger nor more chromatic than hard-edge induced ripples in theFresnel approximation. This enables a broad spectral bandwidth. Notethat in order to better illustrate the oscillations' chromatic naturethis figure has been generated with a large separation between the DMs,Z=2 m. For more realistic separations ACAD exhibits a weaker chromaticbehavior, as illustrated by FIG. 2. Here we intentionally omit thecoronagraphic mask and Lyot stop to highlight that ACAD can operate withany coronagraph.

FIG. 27 shows ACAD in the SR-Fresnel diffractive optics regime+RA-VVC}:We demonstrate a simulated 30% bandwidth contrast of ˜3×10⁹ and an innerworking angle of 2 λ/D on the AFTA pupil using ACAD with two 1 inch DMsseparated by 0.5 m. It demonstrates the weak chromaticity of ACAD andits ability to operate with a high throughput, small IWA coronagraph onan obscured aperture.

DETAILED DESCRIPTION

Some embodiments of the current invention are discussed in detail below.In describing embodiments, specific terminology is employed for the sakeof clarity. However, the invention is not intended to be limited to thespecific terminology so selected. A person skilled in the relevant artwill recognize that other equivalent components can be employed andother methods developed without departing from the broad concepts of thecurrent invention. All references cited anywhere in this specification,including the Background and Detailed Description sections, areincorporated by reference as if each had been individually incorporated.

The terms “light” and “optical” are intended to have broad meanings toinclude both visible and non-visible regions of the electromagneticspectrum. Visible, infrared, ultraviolet and x-ray regions of theelectromagnetic spectrum are all intended to be included within thedefinition of the term “light”.

FIG. 1A is a schematic illustration of an active optical beam shapingsystem 100 according to an embodiment of the current invention. Theoptical beam shaping system 100 includes a first deformable mirror 102arranged to at least partially intercept an entrance beam of light 104and to provide a first reflected beam of light 106, a second deformablemirror 108 arranged to at least partially intercept the first reflectedbeam of light 106 from the first deformable mirror 102 and to provide asecond reflected beam of light 110, and a signal processing and controlsystem 112 configured to communicate with the first and seconddeformable mirrors (102, 108). The signal processing and control system112 is configured to provide control signals to the first deformablemirror 102 so as to configure a reflecting surface of the firstdeformable mirror 102 to substantially conform to a calculated surfaceshape so that the first reflected beam of light 106 will have apredetermined amplitude profile. The signal processing and controlsystem 112 is further configured to provide control signals to thesecond deformable mirror 108 to configure a reflecting surface of thesecond deformable mirror 108 to substantially conform to a calculatedsurface shape so that a phase delay profile across the second reflectedbeam of light 110 will be substantially constant when a phase delayprofile across the entrance beam of light 104 is substantially constant.When the phase delay of entrance beam of light 104 is not constant thesignal processing and control system 112 can control the seconddeformable mirror 108 to make the phase delay of the second reflectedbeam of light 110 constant. The first deformable mirror 102, the seconddeformable mirror 108 and the signal processing and control system 112together provide a large amplitude light modulation range to provide anactively shaped optical beam.

The term “deformable mirror” refers to an assembly that includes areflecting surface as well as structural components that effectreconfiguration of the surface to substantially match calculated surfaceshapes. For example, the reflecting surface can be reconfigured into aplurality of complex non-spherical, non-parabolic and non-ellipsoidalsurface shapes that can include a plurality of local convex and/orconcave localized regions. The structural components of the deformablemirrors can include, but are not limited to, actuators.

The signal processing and control system 112 can be anapplication-specific and/or a general programmable system. For example,it can be implemented on a computer system and/or implemented ondedicated hardware such as an ASIC and/or FPGA. The signal processingsystem and control 112 can also include memory and/or data storage inaddition to data processing components, such as, but not limited to aCPU and/or GPU. The signal processing and control functions can beincorporated together or performed on separate components, for example.The communication paths with the deformable mirrors can be hard wired,such as electrically wired or fiber optic, and/or wireless such as radiofrequency (RF) and/or optical communications links, for example.

In an embodiment of the current invention, the large amplitudemodulation range can be at least 1% of a fully illuminated portion ofthe entrance beam 104 of light. In an embodiment of the currentinvention, the large amplitude modulation range can be substantially anentire range to permit selectable amplitude modulation of portions ofthe entrance beam 104 of light anywhere from substantially fullyattenuated to substantially fully illuminated relative to fullyilluminated portions of the entrance beam of light 104. If the spatialscale amplitude or phase structures of interest of the incoming beam is(δx,δy) ˜O (δ) (where δ is that distance between actuators, OR size ofdiscontinuities between telescope pupil, OR solid angle of a fiber whencoupling to a fiber, or scale of beam/over what info is encoded forlaser can, or scale of beam perturbations), then the large amplitudemodulation regime over this characteristic scale can be defined asfollows:

-   -   Define

$\frac{\partial h}{\partial\overset{\rightarrow}{r}}$

the focal gradient of both mirror's surface at any arbitrary point oneither the input or the output beam (FIG. 1H).

-   -   Define θ_(s) as the solid angle subtended by the disk or radius        δ and the distance z between the DMs.

The radius δ is a characteristic scale of the beam that is sought to bereshaped, Z is a distance between the deformable mirrors, defined by a)the two points for which the optical surfaces of the DMs are thefarthest one to another in the horizontal direction, and b) the lengthbetween the two tangent planes to the DM surfaces at those points. h₁and h₂ are the surface deflections of each DM measured as the relativedistance from each respective DM surface at the tangent planes atmaximum horizontal length defined above.

Then the long amplitude modulation regime is defined when significantbeam remapping occurs.

$\begin{matrix}{{\frac{\partial h}{\partial\overset{\rightarrow}{r}}} \geq {0\left( \theta_{s} \right)}} & {{Eqn}\mspace{14mu} (1)}\end{matrix}$

This rigorous expression can be translated in terms of the largestamplitude modulation (“a”) in a simple case.

Assume we are seeking a 1D amplitude modulation:

$1 + {a\; {{Cos}\left( {\frac{2\; \pi}{D}{nx}} \right)}}$

Then under the assumption a<<1 one can seek:

${h_{1}(x)} = {{- {h_{2}(x)}} = {\varepsilon \; {{Cos}\left( {\frac{2\; \pi}{D}{nx}} \right)}}}$with $\varepsilon = {\frac{n^{2}}{4\; \pi^{2}}\frac{D^{2}}{2}a}$

When in the “large amplitude modulation” regime the “a<<1” breaks downand, this simple solution is not valid, we have to solve theMonge-Ampere equation as per the present description.

Assuming that the characteristic scale of the input beam is

$\frac{D}{N_{act}}$

(actuator pitch) and that n is “as large as possible” in order not tolose generality, eg,

$n = {\frac{N_{act}}{2}.}$

Then folding this into Eqn (1) yields:

$a_{crit} = {\sigma \left( \frac{4\; \pi}{N_{act}} \right)}$

Namely, when N=32 we find a_(crit)=1.2% as previously stated in thisspecification, the boundary between “small” and “large” amplitudemodulation is around 1%, but the above also provides a rigorousderivation. This is the rigorous definition that defines the boundarybetween small and large amplitude modulation regime for ALLapplications.

In an embodiment of the current invention, the active optical beamshaping system 100 can further include a beam analyzer 114 arranged toreceive at least a portion of the exit beam 110 of light reflected fromthe second deformable mirror 108. In alternative embodiments, the beamanalyzer 114 can receive external information regarding the medium orenvironment in which the second beam of light 110 will propagate. Thebeam analyzer 114 can be configured to communicate with the signalprocessing and control system 112 to provide feedback control of atleast one of the first and second deformable mirrors (102, 108). Thebeam analyzer 114 can be hard wired by electrical and/or fiber opticconnections to the signal processing and control system 112, and/or inwireless communication such as by RF and/or optical wirelesscommunications links. In some embodiments, the beam analyzer 114 can befurther configured to estimate: (1) the amplitude profile of the beamand/or (2) the phase of the beam.

In an embodiment of the current invention, input 116 for a desiredoutput beam shape can be provided to the signal processing and controlsystem 112. The input 116 information for the desired output beam shapecan be used in conjunction with feedback signals from said beam analyzer114 in some embodiments of the current invention. However, embodimentsin which no feedback information is used are intended to be includedwithin the general concepts of the current invention.

In an embodiment of the current invention, the signal processing andcontrol system 112 can be further configured to calculate the surfaceshape of at least one of the first and second deformable mirrors (102,108) based on reverse ray tracing calculations.

In an embodiment of the current invention, the signal processing andcontrol system 112 can be further configured to calculate the surfaceshape of at least one of the first and second deformable mirrors basedon a solution to Monge-Ampere equations corresponding to the activeoptical beam shaping system.

Further embodiments of the current invention are directed to numerousdevices and systems that incorporate at least one active optical beamshaping system according to an embodiment of the current invention. Thefollowing will describe some such devices and systems; however, oneshould recognize that further devices and systems, and modificationsthereof, can be constructed with one or more active optical beam shapingsystems according to an embodiment of the current invention and areconsidered to be included within the general scope of the currentinvention.

FIG. 1B is a schematic illustration of an optical transmitter 200according to an embodiment of the current invention. The opticaltransmitter 200 includes an optical source 202 and an optical modulator204 arranged in an optical path 206 of the optical source 202. Theoptical modulator 204 can be similar to or the same as the activeoptical beam shaping system 100, for example. The optical modulator 204includes, similar to FIG. 1A and not shown in FIG. 1B, a firstdeformable mirror arranged to at least partially intercept an entrancebeam of light from the optical source 202 and to provide a firstreflected beam of light, a second deformable mirror arranged to at leastpartially intercept the first reflected beam of light from said firstdeformable mirror and to provide a second reflected beam of light, and asignal processing and control system configured to communicate with thefirst and second deformable mirrors. The signal processing and controlsystem is configured to provide control signals to the first deformablemirror and the second deformable mirror to provide at least a portion ofthe second reflected beam of light as an output beam 207 having at leastone of a selectable amplitude profile or selectable phase profile.

The optical transmitter 200 can further include a beam analyzer 208configured to sample a portion of the output beam 207. For example, theoptical transmitter 200 can include a beam splitter 210, such as apartially silvered mirror or a dichroic mirror arranged in the outputbeam 207 to redirect a portion of the output beam 207 to beam analyzer208. However, the general concepts of the current invention are notlimited to this particular example. The beam analyzer can be, but is notlimited to, a camera and/or a wavefront sensor, for example.

The optical transmitter 200 can be used in, but is not limited to,optical communications systems for example. Such optical communicationssystems can be free space systems, or can be waveguide systems, forexample. Waveguide systems can include, but are not limited to, fiberoptic systems. The term “free space” is intended to have a generalmeaning to include vacuum propagation, such as between satellites and/orspacecraft, or through a medium such as the atmosphere, and/or water.

In some embodiments, the signal processing and control system of theoptical transmitter 200 can be configured to provide control signals tothe first deformable mirror and the second deformable mirror such thatthe output beam has both a selectable amplitude profile and selectablephase profile. In some embodiments, the signal processing and controlsystem of the optical transmitter 200 can be configured to providecontrol signals to the first deformable mirror and the second deformablemirror such that the output beam of light 207 is soliton pulses. Thesoliton boundary conditions can be input to the signal processing andcontrol system, for example. For a given fiber we do not calculate thesoliton shape here. However, we can couple to the fiber any prescribedsolitons and keep accurately their profile in the presence ofdisturbances.

The solitons can be coupled into an optical fiber, for example, for longdistance fiber optic communication without the need for repeaters forsignal regeneration, for example. This can be useful for transoceanicfiber optic systems, for example.

FIG. 1C is a schematic illustration of laser beam launch system 220according to an embodiment of the current invention. It includes anoptical transmitter 222 and reflecting telescope 224. The opticaltransmitter 222 can be the same or similar to optical transmitter 200,for example. Some applications of laser beam launch system 220 caninclude, but are not limited to, tactical and strategic shooting down ofsatellites.

FIG. 1D is a schematic illustration of an optical receiver 300,according to an embodiment of the current invention. The opticalreceiver 300 includes an optical detector 302 and an optical filter 304arranged in an optical path of the optical detector 320. The opticalfilter can be similar to or the same as the active optical beam shapingsystem 100, for example. The optical filter 304 includes a firstdeformable mirror arranged to at least partially intercept an entrancebeam of light being detected to provide a first reflected beam of light,a second deformable mirror arranged to at least partially intercept thefirst reflected beam of light from the first deformable mirror and toprovide a second reflected beam of light, and a signal processing andcontrol system configured to communicate with the first and seconddeformable mirrors, (Also, see, e.g., FIG. 1A.) The signal processingand control system is configured to provide control signals to the firstdeformable mirror and the second deformable mirror to provide at least aportion of the second reflected beam of light as an output beam directedto the optical detector to have at least one of a selectable amplitudeprofile or selectable phase profile.

The optical receiver 300 can further include a beam analyzer 306configured to communicate with the optical filter 304. The beam analyzer306 can be, but is not limited to, a camera and/or wavefront detector,for example. The optical receiver 300 can further include a beamsplitter 308, such as, but not limited to, a partially silvered mirroror a dichroic mirror to direct a portion of the received beam of lightto the beam analyzer. The optical receiver 300 can be used in, but isnot limited to, an optical communication system. The opticalcommunication system can be a waveguide system and/or a free spacesystem.

In free space communication systems according to some embodiments of thecurrent invention, information can be encoded in the shape of the beamwithout loss of light, such as would occur with temporal modulation. Thebroad bandwidth can be advantageous for communications.

FIG. 1E is a schematic illustration of a solar furnace 400 according toan embodiment of the current invention. The solar furnace 400 includes alight collection system 402, an active optical beam shaper 404 arrangedin an optical path of light collected by the light collection system402, and an optical focusing system 406 arranged to receive a beam oflight from the active optical beam shaper 404 to be focused on an object408 to be heated. The active optical beam shaper 404 can be the same as,or similar to, active optical beam shaping system 100. The efficiency isboosted by shaping the solar beam precisely to the heat element withlosing energy that can scatter and also cause damage.

The following examples describe some embodiments and some applicationsin more detail. However, the broad concepts of the current invention arenot limited to the particular examples.

EXAMPLES

The examples here describe a family of practical applications accordingto some embodiments of the current invention. As their ultimateperformances depend strongly on the pupil structure, we limit the scopehere to a few characteristic examples. Full optimization for specifictelescope geometries, as well as other applications, can be conducted asneeded.

1 Introduction

Some embodiments of the current invention can take advantage ofstate-of-the-art Deformable Mirrors (DM) in modern high-contrastinstruments to address the problem of pupil amplitude discontinuitiesfor on-axis and/or segmented telescopes. Indeed, coronagraphs are notsufficient to reach the high contrast required to image faintexo-planets: wavefront control is needed to remove the light scatteredby small imperfections on the optical surfaces (Brown & Burrows 1990).Over the past several years, significant progress has been made in thisarea, both in the development of new algorithms (Bordé & Traub 2006;Give'on et al. 2007) and in the experimental demonstration ofhigh-contrast imaging with a variety of coronagraphs (Give'on et al.2007; Trauger & Traub 2007; Guyon et al. 2010b; Belikov et al, 2011).These experiments rely on a system with a single Deformable Mirror whichis controlled based on diagnostics downstream of the coronagraph, eitherat the science camera or as close as possible to the end detector(Wallace et al. 2009; Pueyo et al. 2010). Such configurations are wellsuited to correct phase wavefront errors arising from surface roughnessbut have limitations in the presence of pure amplitude errors(reflectivity), or phase-induced amplitude errors, which result from thepropagation of surface errors in optics that are not conjugate to thetelescope pupil (Shaklan & Green 2006; Pueyo & Kasdin 2007). Indeed asingle DM can only mimic half of the spatial frequency content ofamplitude errors and compensate for them only on one half of the imageplane (thus limiting the scientific field of view) over a moderatebandwidth. In theory, architectures with two sequential DeformableMirrors, can circumvent this problem and create a symmetric broadbandhigh contrast PSF (Shaklan & Green 2006; Pueyo & Kasdin 2007). The firstdemonstration of symmetric dark hole was reported in Pueyo et al. (2009)and has since been generalized to broadband by Groff et al. (2011). Insuch experiments the coronagraph has been designed over a full circularaperture, the DM control strategy is based on a linearization of therelationship between surface deformations and electrical field at thescience camera, and the modeling tools underlying the control loopconsist of classic Fourier and Fresnel propagators. This is illustratedon the left panel of FIG. 1. As a consequence, a wavefront controlsystem composed of two sequential Deformable Mirrors is currently thebaseline architecture of currently envisioned coronagraphic space-basedinstruments (Shaklan et al. 2006; Krist et al. 2011) and ELT planetimagers (Macintosh et al. 2006). One can thus naturally be motivated toinvestigate if such wavefront control systems can be used to cancel thelight diffracted by secondary supports and segments in large telescopes,since such structures are amplitude errors, albeit large amplitudeerrors.

The purpose of the current example is to demonstrate that indeed a twoDeformable Mirror wavefront control system can mitigate the impact ofthe pupil asymmetries, such as spiders and segments, on contrast andthus enable high contrast on unfriendly apertures. We first present anew approach to coronagraph design according to an embodiment of thecurrent invention in the presence of a central obscuration, but in theabsence of spiders or segments. We show that for coronagraphs with apupil apodization and an opaque focal plane stop, contrasts of 10⁻¹⁰ canbe reached for any central obscuration diameter, provided that the InnerWorking Angle is large enough. Naturally the secondary supportstructures, and in the segmented cases, segment gaps, will degrade thiscontrast. As our goal is to use two DMs as an amplitude modulationdevice, we first briefly review the physics of such a modulation. Wethen introduce a solution to this problem. In particular, we show how tocompute DM surfaces that mitigate spiders and segment gaps. Currentalgorithms used for amplitude control operate under the assumption thatamplitude errors are small, and thus they cannot be readily applied tothe problem of compensating aperture discontinuities, which haveinherently large reflectivity non-uniformities. FIG. 1G illustrates howthe current example embodiment introduces a control strategy for the DMsthat is radically different from previously published amplitudemodulators in the high-contrast imaging literature. Our technique, whichwe name Active Compensation of Aperture Discontinuities (hereafterACAD), finds the adequate DM shapes in the true non-linear largeamplitude error regime. In this case the DMs' surfaces are calculated asthe solution of a non-linear partial differential equation, called theMonge-Ampere Equation. Below, we further describe our methodologyaccording to an embodiment of the current invention to solve thisequation and illustrate each step using an obscured and segmentedgeometry similar to JWST. As ACAD DM surfaces are prescribed in theray-optics approximation, this is a fundamentally broadband techniqueprovided that chromatic diffractive artifacts, edge ringing inparticular, do not significantly impact the contrast. We find that whenremapping small discontinuities with Deformable Mirrors, the spectralbandwidth is only limited by wavelength-dependent edge-diffractionringing in the Fresnel approximation (as discussed in Pueyo & Kasdin(2007) for instance). Below, we also present the application of someembodiments of the current invention to various observatoryarchitectures. Some embodiments of the current invention can render highcontrast coronagraphy possible on any observatory geometry withoutadding any new hardware.

2 Coronagraphy with a Central Obscuration

2.1 Optimizing Pupil Apodization in the Presence of a CentralObscuration

Because the pupil obscuration in an on-axis telescope is large it willbe very difficult to mitigate its impact with DMs with a limited stroke.Indeed, the main hindrance to high-contrast coronagraphy in on-axistelescope is the presence of the central obscuration: it often shadowsmuch more than 10% of the aperture width while secondary supports andsegments gaps cover ˜1%. We thus first focus of azimuthally symmetriccoronagraphic designs in the presence of a central obscuration. Thisproblem (without the support structures) has been addressed in previouspublications either using circularly symmetric pupil apodization(Soummer et al. 2011) or a series of phase masks (Mawet et al. 2011).Both solutions however are subject to limitations. The singularity atthe center of the Optical Vector Vortex Coronagraph might be difficultto manufacture and a circular opaque spot thus lies in the centralportion of the phase mask (<λ/D) which results in a degradation of theideal contrast of such a coronagraph (Krist et al. 2011). The solutionsin Soummer et al. (2011) result from an optimization seeking to maximizethe off-axis throughput for a given focal plane stop diameter: the finalcontrast is absent from the optimization metric and is only a byproductof the chosen geometry. Higher contrasts are then obtained by increasingthe size of the focal plane mask, and thus result in a loss in IWA.

2.2 The Optimization Problem

Here we revisit the solution proposed by Soummer et˜al. (2011) in aslightly different framework. We recognize that, in the presence ofwavefront errors, high contrast can only be achieved in an area of thefield of view that is bounded by the spatial frequency corresponding tothe DM's actuator spacing. We thus consider the design of an ApodizedPupil Lyot Coronagraph which only aims at generating high contrastbetween the Inner Working Angle (IWA) and Outer Working Angle (OWA). Inorder to do so, we rewrite coronagraphs described by Soummer et˜al.(2003) as an operator

which relates the entrance pupil P(r) to the electrical field in thefinal image plane. We first call {circumflex over (P)}(ξ) the Hankeltransform of the entrance pupil:

{circumflex over (P)}(ξ)=∫_(D) _(S) _(/2) ^(D/2) P(r)J ₀(rξ)rdr   (1)

where D is the pupil diameter, D_(S) the diameter of the secondary and ξthe coordinate at the science detector expressed in units of angularresolution (λ₀/D). λ₀ is the design wavelength of the coronagraph chosento translate the actual physical size of the focal plane mask in unitsof angular resolution (often at the center of the bandwidth ofinterest). λ is then the wavelength at which the coronagraph isoperating (e.g. the physical size of the focal plane mask remainsconstant as the width of the diffraction pattern changes withwavelength). For the purpose of the monochromatic designs presentedherein λ=λ₀. Then the operator is given by:

$\begin{matrix}{{{\left\lbrack {P(r)} \right\rbrack}(\xi)} = {{\hat{P}(\xi)} - {\frac{\lambda^{2}}{\lambda_{0}^{2}}{\int_{D_{S}/2}^{D/2}{{\hat{P}(\eta)}{K\left( {\xi,\eta} \right)}\eta \ {\eta}}}}}} & (2)\end{matrix}$

where K(ξ,η) is the convolution kernel that captures the effect of thefocal plane stop of diameter M_(stop):

K(ξ,η)=∫₀ ^(M) ^(stop) ^(/2) J ₀(uη)J ₀(uξ)udu   (3)

An analytical closed form for this kernel can be calculated using Lommelfunctions. Note that this Eq. 2 assumes that the Lyot stop is notundersized. Since we are interested in high contrast regions that onlyspan radially all the way up to a finite OWA, we seek pupil apodizationof the form:

$\begin{matrix}{{P(r)} = {\sum\limits_{k = 0}^{N_{modes}}\; {p_{k}{J_{Q}\left( \frac{r}{\alpha_{k}^{Q}} \right)}}}} & (4)\end{matrix}$

where I_(Q)(r) denotes the Bessel function of the first kind of order Qand α_(k) ^(Q) the kth zero of this Bessel function. In order to deviseoptimal apodizations over obscured pupils, Q, can be chosen to be largeenough so that I_(Q)(r)<<1 for r<D_(S)/2 (in practice we choose Q=10),The α_(k) ^(Q) corresponds to the spatial scale of oscillations in thecoronagraph entrance pupil, and such a basis set yields high contrastregions all the way to OWA≅N_(modes)λ₀/D. Since the operator in Eq. 2 islinear, finding the optimal P_(k) can be written as the following linearprogramming problem:

$\begin{matrix}{\max\limits_{(p_{k})}\; {\left\lbrack {\min\limits_{r}\left( {P(r)} \right)} \right\rbrack \mspace{14mu} {Under}\mspace{14mu} {the}\mspace{14mu} {constraints}\text{:}}} & (5) \\{{{{\left\lbrack \left( {P(r)} \right) \right\rbrack}(\xi)}} < {10^{- \sqrt{C}}\mspace{14mu} {for}\mspace{14mu} {IWA}} < \xi < {OWA}} & (6) \\{{\max\limits_{r}\left( {P(r)} \right)} = 1} & (7) \\{{{\frac{\;}{r}\left\lbrack {P(r)} \right\rbrack}} < {b.}} & (8)\end{matrix}$

Our choice of cost function and constraints has been directed by thefollowing rationale:

-   -   We maximize the smallest value on the apodization function in an        attempt to maximize throughput. The actual throughput is a        quadratic function of the p_(k). Maximizing it requires the        solution of a non-linear optimization problem (as described in        Vanderbei et˜al. (2003).    -   The contrast constraint is enforced between the IWA and the OWA        (<N_(modes)).    -   The maximum of the apodization function is set to one (otherwise        the p_(k) will be chosen to be sufficiently small so that the        contrast constraint is met).    -   The absolute value of the derivative across the pupil cannot be        larger than a limit, denoted as b here. As the natural solutions        of such problems are very oscillatory (or “bang bang”,        [Vanderbei et˜al, (2003)], a smoothness constraint has to be        enforced (see [Vanderbei et˜al. (2007)] for a similar case).

Note that the linear transfer function in Eq. 2 can also be derived forother coronagraphs, with grayscale and phase image-plane masks, or forthe case of under-sized Lyot stops. As general coronagraphic design inobscured circular geometries is not our main purpose, we limit the scopeof the paper to coronagraphs represented by Eq. 2.

Optimal design Apodized Pupil Lyot Coronagraphs on circularly obscuredapertures. With fixed obscuration ratio, size of opaque focal planemask, IWA and OWA, our linear programming approach yields solutions withtheoretical contrast below 10¹⁰. All PSFs shown on this figure aremonochromatic for λ=λ₀. When all other quantities remain equal and thecentral obscuration ratio increases (from 10% in the top panel to 20% inthe middle panel), then the solution becomes more oscillatory (e.g lessfeasible) and the contrast constraint has to be relaxed. Eventually theoptimizer does not find a solution and the size of the opaque focalplane mask (and thus the IWA) has to be increased (central obscurationof 30% in the bottom panel). On the right hand side, we present ourresults in two configurations: when the apodization is achieved using atgrayscale screen (APLC), “on-sky” λ/D bottom x-axis, and when theapodization is achieved via two pupil remapping mirrors (PIAAC),“on-sky” λ/D top x-axis. We adopt this presentation to show that ACAD is“coronagraph independent” and that it can be applied to coronagraphswith high throughput and small IWA.

2.3 Results of the Optimization

Typical results of the monochromatic optimization in Eqs. 8, with λ=λ₀,are shown in FIG. 2.2 for central obscurations of 10, 20 and 30%. In thefirst two cases the size of the focal plane stop is equal to 3λ/D, theIWA is 4λ/D and the OWA is 30λ/D. As the size of the central obscurationincreases the resulting optimal apodization becomes more oscillatory andthe contrast constraint has to be loosened in order for the linearprogramming optimizer to converge to a smoother solution. Alternativelyincreasing the size of the focal plane stop yields smooth apodizers withhigh contrast, at a cost in angular resolution (bottom panel with acentral obscuration of 30%, a focal plane mask of radius 4λ/D, an IWA of5λ/D and an OWA of 30λ/D). These trade-offs were described in Soummeret˜al. (2011), however our linear programming approach to the design ofpupil apodizations now imposes the final contrast instead of having itbe a by product of fixed central obscuration and focal plane stop. Theseapodizations can either be generated using grayscale screen (at a costin throughput and angular resolution) or a series of two aspherical PIAAmirrors (for better throughput and angular resolution). In order not tolose generality, we present our results on FIG. 2.2 considering the twotypes of practical implementations (classical apodization and PIAAapodization). In the case of a grayscale amplitude screen the angularresolution units are as defined in Eq. 2 and the throughput is smallerthan unity. In the case of PIAA apodisation the throughput is unity andthe angular resolution units have been magnified by the fieldindependent centroid based angular magnification defined in Pueyo et˜al.(2011). We adopt this presentation for the remainder of the paper whereone dimensional PSFs will be presented with “APLC angular resolutionunits” in the bottom horizontal axis and“PIAAC angular resolution units”in the top horizontal axis.

Note that this linear programming approach only optimizes the contrastfor a given wavelength. However, since the solutions presented in FIG.2.2 feature contrasts below 10¹⁰, we choose not to focus on coronagraphchromatic optimizations. Instead, in order to account for the chromaticbehavior of the coronagraph, the monochromatic simulations in §. 6 arecarried out under the conservative assumption that the physical size ofthe focal plane stop is somewhat smaller than optimal (or that theoperating wavelength of the coronagraph is slightly off, λ=1.2λ₀). As aconsequence the raw contrast of the coronagraphs presented in §. 6 is˜10⁻⁹ . Note that this choice is not representative of all possibleApodized Pupil Coronagraph chromatic configurations. It is merely ashortcut we use to cover the variety of cases presented in §. 6. In §.7.2 we present a set of broadband simulations that include wavefronterrors and the true coronagraphic chromaticity for a specificconfiguration and show that bandwidth is more likely to be limited bythe spectral bandwidth of the wavefront control system than by thecoronagraph. However, future studies aimed at defining the true contrastlimits of a given telescope geometry will have to rely on solutions ofthe linear problem in Eqs. 8 which has been augmented to accommodate forbroadband observations. In theory, the method presented here can also beapplied to asymmetric pupils. However, the optimization quickly becomescomputationally intensive as the dimensionality of the linearprogramming increases (in particular when the smoothness constraint andthe bounds on the apodization have to be enforced at all points of a twodimensional array). This problem can be somewhat mitigated when seekingfor binary apodizations, as shown in Carlotti et˜al. (2011), at a costin throughput and angular resolution.

3 Physics of Amplitude Modulation 3.1 General Equations

We have shown in §. 2 that by considering the design of pupil apodizedcoronagraphs in the presence of a circular central obscuration as alinear optimization problem, high contrast can be reached provided thatthe focal plane mask is large enough. In practice, the secondary supportstructures and the other asymmetric discontinuities in the telescopeaperture (such as segment gaps) will prevent such levels of starlightsuppression. We demonstrate that well controlled DMs can circumvent theobstacle of spiders and segment gaps. In this section, we first set-upour notations and review the physics of phase to amplitude modulation.We consider the system represented on FIG. 3 where two sequential DMsare located between the telescope aperture and the entrance pupil of thecoronagraph. In this configuration, the telescope aperture and the pupilapodizer are not in conjugate planes. This will have an impact on thechromaticity of the system and is discussed in §. 5. Without loss ofgenerality we work under the “folded” assumption illustrated on FIG. 3where the DMs are not tilted with respect to the optical axis and can beconsidered as lenses of index of refraction −1 (as discussed in[Vanderbei Traub(2005)]. In the scalar approximation the relationshipbetween the incoming field, E_(DM1)(x,y), and the outgoing field,E_(DM2)(x₂,y₂), is given by the diffractive Huygens Integral:

$\begin{matrix}{{E_{{DM}\; 2}\left( {x_{2},y_{2}} \right)} = {\frac{1}{\; \lambda \; Z}{\int_{}^{\;}{{E_{{DM}\; 1}\left( {x,y} \right)}^{\frac{2\; \pi}{\lambda}{Q{({x,y,x_{2},y_{2}})}}}\ {x}{y}}}}} & (9)\end{matrix}$

where

corresponds to the telescope aperture and Q(x,y,x₂,y₂) stands for theoptical path length between any two points at DM1 and DM2:

Q(x,y,x ₂ ,y ₂)=h ₁(x,y)+S(x,y,x ₂,y₂)−h ₂(x ₂ ,y ₂)   (10)

S(x,y,x₂,y₂) is the free space propagation between the DMs:

$\begin{matrix}\sqrt{\left( {x - x_{2}} \right)^{2} + \left( {y - y_{2}} \right)^{2} + \left( {Z + {h_{1}\left( {x,y} \right)} - {h_{2}\left( {x_{2},y_{2}} \right)}} \right)^{2}} & (11)\end{matrix}$

where Z is the distance between between the two DMs, h₁ and h₂ are theshapes of DM1 and DM2 respectively (as shown on FIG. 3) and λ is thewavelength. We recognize that two sequential DMs act as a pupilremapping unit similar to PIAA coronagraph [Guyon(2003)] whose rayoptics equations were first derived by [Traub & Vanderbei(2003)]. Webriefly state the notation used to describe such an optical system asintroduced in Pueyo et˜al. (2011):

For a given location at DM2, (x₂,y₂), the location at DM1 of theincident ray according to ray optics is given by:

x ₁(x ₂ ,y ₂)=f ₁(x ₂ ,y ₂)   (12)

y ₁(x ₂ ,y ₂)=g ₁(x ₂ ,y ₂).   (13)

Conversely, for a given location at DM1, (x₁,y₁), the location at DM2 ofthe outgoing ray according to geometric optics is given by:

x ₂(x ₁ ,y ₁)=f ₂(x ₁ ,y ₁)   (14)

y ₂(x ₁ ,y ₁)=g ₂(x ₁ ,y ₁).   (15)

Fermat's principle dictates the following relationships between theremapping functions and the shape of DM1:

$\begin{matrix}{{\frac{\partial h_{1}}{\partial y}_{({x_{1},y_{1}})}} = \frac{x_{1} - {f_{2}\left( {x_{1},y_{1}} \right)}}{Z}} & (16) \\{{\frac{\partial h_{1}}{\partial y}_{({x_{1},y_{1}})}} = {\frac{y_{1} - {g_{2}\left( {x_{1},y_{1}} \right)}}{Z}.}} & (17)\end{matrix}$

Conversely, if we choose the surface of DM2 to ensure that the outgoingon-axis wavefront is flat, we then find:

$\begin{matrix}{{\frac{\partial h_{2}}{\partial x}_{({x_{2},y_{2}})}} = \frac{x_{2} - {f_{1}\left( {x_{2},y_{2}} \right)}}{Z}} & (18) \\{{\frac{\partial h_{2}}{\partial y}_{({x_{2},y_{2}})}} = {\frac{y_{2} - {g_{1}\left( {x_{2},y_{2}} \right)}}{Z}.}} & (19)\end{matrix}$

Schematic of the optical system considered: the telescope apertures isfollowed by two sequential Defromable Mirrors (DMs) in non-conjugateplanes whose purpose is to remap the pupil discontinuities. The beamthen enters a coronagraph to suppress the bulk of the starlight: in thisfigure we show an Apodized Lyot Pupil Coronagraph (APLC). This is thecoronagraphic architecture we consider for the remainder of the paperbut we stress that the method presented herein is applicable to anycoronagraph.

3.2 Fresnel Approximation and Talbot Imaging

In Pueyo et˜al. (2011) we showed that one could approximate thepropagation integral in Eq. 9 by taking in a second order Taylorexpansion of Q(x,y,x₂,y₂) around the rays that trace (f₁(x₂,y₂),g₁(x₂,y₂)) to (x₂,y₂). In this case the relationship between the fieldsat DM1 and DM2 is:

$\begin{matrix}{{E_{{DM}\; 2}\left( {x_{2},y_{2}} \right)} = {{\frac{^{\frac{z\; {\pi}}{\lambda}Z}}{{\lambda}\; Z}\left\{ {\int_{}{{E_{{DM}\; 1}\left( {x,y} \right)}^{\frac{\pi}{\lambda \; Z}{\lbrack{{\frac{\partial f_{2}}{\partial x}{({x - x_{1}})}^{2}} + {2\frac{\partial g_{2}}{\partial x}{({x - x_{1}})}{({y - y_{1}})}} + {\frac{\partial g_{2}}{\partial y}{({y - y_{1}})}^{z}}}\rbrack}}{x}{y}}} \right\}}_{({x_{1},y_{1}})}}} & (20)\end{matrix}$

When the mirror's deformations are very small compared to both thewavelength and D²/Z, the net effect of the wavefront disturbance createdby DM1 can be captured in E_(DM1)(x,y) and the surface of DM2 can befactored out of Eq. 9. In this case

${x_{1} = x_{2}},{y_{1} = y_{2}},{{\frac{\partial f_{2}}{\partial x}_{x_{1},y_{1}}} = 1},{{\frac{\partial f_{2}}{\partial y}_{x_{1},y_{1}}} = 0},{{\frac{\partial g_{2}}{\partial y}_{x_{1},y_{1}}} = 1.}$

Then, Eq. 20 reduces to:

$\begin{matrix}{{E_{{DM}\; 2}\left( {x_{2},y_{2}} \right)} = {\frac{^{\frac{2\; {\pi}}{\lambda}{({Z - {h_{2}{({x_{2},y_{2}})}}})}}}{{\lambda}\; Z}{\int_{Aperture}{^{\frac{2\; {\pi}}{\lambda}{({h_{1}{({x,y})}})}}^{\frac{\pi}{\lambda \; Z}{({{({x - x_{2}})}^{z} + {({y - y_{2}})}^{2}})}}{x}{y}}}}} & (21)\end{matrix}$

which is the Fresnel approximation. If moreover

${{h_{1}\left( {x,y} \right)} = {{\lambda ɛcos}\left( {\frac{2\pi}{D}\left( {{mx} + {ny}} \right)} \right)}},$

h₂(x,y)=−h₁(x,y), with ε<<1, then the outgoing field is to first order:

$\begin{matrix}{{{E_{{DM}\; 2}\left( {x_{2},y_{2}} \right)} \propto {\frac{{\pi\lambda}\; {Z\left( {m^{2} + n^{2}} \right)}}{D^{2}}{{{\lambda ɛcos}\left( {\frac{2\pi}{D}\left( {{mx}_{2} + {ny}_{2}} \right)} \right)}.{E_{{DM}\; 2}\left( {x_{2},y_{2}} \right)}}}} = {\frac{n\; \lambda \; {Z\left( {m^{2} + n^{2}} \right)}}{p^{2}}{{\lambda\varepsilon cos}\left( {\frac{2\pi}{p}\left( {{mx}_{2} + {ny}_{2}} \right)} \right)}}} & (22)\end{matrix}$

This phase-to-amplitude coupling is a well known optical phenomenoncalled Talbot imaging and was introduced to the context of high contrastimaging by [Shaklan & Green (2006)]. In the small deformation regime,the phase on DM1 becomes an amplitude at DM2 according to the couplingin Eq. 22. When two sequential DMs are controlled to cancel smallamplitude errors, as in Pueyo et˜al. (2009), they operate in thisregime. Note, however, that the coupling factor scales with wavelength(the resulting amplitude modulation is wavelength independent, but thecoupling scales as λ): this formalism is thus not applicable to ourcase, for which we are seeking to correct large amplitude errors(secondary support structures and segments) with the DMs. In practice,when using Eq. 22 in the wavefront control scheme outlined in Pueyoet˜al. (2009) to correct aperture discontinuities, this weak couplingresults in large mirror shapes that lie beyond the range of the linearassumption made by the DM control algorithm. For this reason, methodsoutlined on the left panel of FIG. 1G to correct for aperturediscontinuities do not converge to high contrast. Because phase toamplitude conversion is fundamentally a very non-linear phenomena, thesedescending gradient methods [Bordà & Traub (2006), Give'on et˜al. (2007)Pueyo et˜al. (2009)] are not suitable to find DM shapes that mitigateapertures discontinuities. We circumvent these numerical limitations bycalculating DMs shapes that are based on the full non-linear problem,right panel of FIG. 1G.

3.3 The SR-Fresnel Approximation

In the general case, starting from Eq. 20 and following the derivationdescribed in §. 5, the field at DM2 can be written as follows:

$\begin{matrix}{{E_{{DM}\; 2}\left( {x_{2},y_{2}} \right)} = {\left\{ {\sqrt{{\det \lbrack J\rbrack}}{\int_{\mathcal{F}}{\left( {\xi,\eta} \right)^{{2\pi}{({{\xi \; f_{1}} + {\eta \; g_{1}}})}}^{{- }\frac{{\pi\lambda}\; Z}{\det {\lbrack J\rbrack}}{({{\frac{\partial g_{1}}{\partial y}\xi^{2}} + {\frac{\partial f_{1}}{\partial x}\eta^{2}}})}}{\xi}{\eta}}}} \right\} _{({x_{2},y_{2}})}}} & (23)\end{matrix}$

where

(ξ,η) is the Fourier transform of the telescope aperture,

and stands for the Fourier plane. We call this integral theStretched-Remapped Fresnel approximation (SR-Fresnel). Moreoverdet[J(x₂,y₂)] is the determinant of the Jacobian of the change ofvariables that maps (x₂,y₂) to (x₁,y₁):

$\begin{matrix}{{\det \left\lbrack {J\left( {x_{2},y_{2}} \right)} \right\rbrack} = {\left\{ {{\frac{\partial f_{1}}{\partial x}\frac{\partial g_{1}}{\partial y}} - \left( \frac{\partial g_{1}}{\partial x} \right)^{2}} \right\} _{({x_{2},y_{2}})}.}} & (24)\end{matrix}$

In the ray optics approximation, λ˜0, the non-linear transfer functionbetween the two DMs becomes:

E _(DM2)(x ₂ ,y ₂)={√{square root over (∥det[J]∥)}E_(DM1)(f ₁ ,g₁)}∥_((x) ₂ _(,y) ₂ ₎   (25)

[E _(DM2)(x ₂ ,y ₂)]² ={det[J][E _(DM1)(f ₁ ,g ₁)]²}∥_((x) ₂ _(,y) ₂ ₎  (26)

The square form (e.g Eq. 26) of this transfer function can also bederived based on conservation of energy principles and is ageneralization to arbitrary geometries of the equation driving thedesign of PIAA coronagraphs [Vanderbei & Traub (2005)]. A fulldiffractive optimization of the DM surfaces requires use of the completetransfer function shown in Eq. 23. However, there do not exist yettractable numerical method to evaluate Eq. 23 efficiently enough inorder for this model to be included in an optimization algorithm.Moreover even solving the ray optics problem is extremely complicated:it requires to find the mapping function (f₁,g₁) which solves thenon-linear partial differential equation in Eq. 26. Substituting for(f₁,g₁) and using Eqs. 19 yields a second order non-linear partialdifferential equation in h₂. This is the problem that we set ourselvesto tackle in the next section, and is the cornerstone of our AdaptiveCompensation of Aperture Discontinuities. As a check, one can verifythat in the small deformation regime

$\left( {{{e.g.\mspace{14mu} {if}}\mspace{14mu} {h_{1}\left( {x,y} \right)}} = {{{{\lambda ɛcos}\left( {\frac{2\pi}{D}\left( {{mx} + {ny}} \right)} \right)}\mspace{14mu} {and}\mspace{14mu} {h_{2}\left( {x,y} \right)}} = {- {h_{1}\left( {x,y} \right)}}}} \right)$

Eq. 26 yields the same phase-to-amplitude coupling as in Talbot imaging[Pueyo (2008)], Eq. 26 is a well know optimal transport problem[Monge(1781)], which has already been identified as underlying opticalillumination optimizations [Glimm & Oliker (2002)]. While the existenceand uniqueness of solutions in arbitrary dimensions have beenextensively discussed in the mathematical literature (see [Dacorogna &Moser (1990)] for a review), there was no practical numerical solutionpublished up until recently. In particular, to our knowledge, not even adimensional solution for which the DM surfaces can be described using arealistic basis-set has been published yet. We now introduce a methodthat calculates solutions to Eq. 26 which can be represented by feasibleDM shapes.

4 Calculation of the Deformable Mirror Shapes 4.1 Statement of theProblem

Ideally, we seek DM shapes that fully cancel all the discontinuities atthe surface of the primary mirror and yield a uniform amplitudedistribution, as shown in the top panel of FIG. 2. A solutions for aparticular geometry with four secondary support has been derived by Loziet˜al. (2009). It relies on reducing the dimensionality of the problemto the direction orthogonal to the spiders. It is implemented using atransmissive correcting plate that is a four-faced prism arranged suchthat the vertices coincides with the location of the spiders. Thecurvature discontinuities at the location of the spiders are responsiblefor the local remapping that removes the spiders in the coronagraphpupil. However such a solution cannot be readily generalized to the caseof more complex apertures, where the secondary support structures mightvary in width, or in the presence of segment gaps. Moreover it istransmissive and thus highly chromatic. Here we focus on a differentclass of solutions and seek to answer a different question. How well canwe mitigate the effect of pupil discontinuities using DMs with smoothsurfaces, a limited number of actuators (e.g a limited maximalcurvature), and a limited stroke? Under these constraints directlysolving Eq. 26 (e.g. solving the forward problem illustrated in the toppanel of FIG. 2) is not tractable as both factors on the left hand sideof Eq. 26 depend on h₂. More specifically, the implicit dependence ofE_(DM1) (f₁(x₂,y₂), g₁(x₂,y₂)) on h₂ can only be addressed using finiteelements solvers, whose solutions might not be realisticallyrepresentable using a DM. However this can be circumvented using thereversibility of light and solving the reverse problem, where the twomirrors have been swapped. Indeed, since

x ₂ =f ₂(f ₁(x ₂ ,y ₂),g ₁(x ₂ ,y ₂))   (27)

y ₂ =g ₂(f ₁(x ₂ ,y ₂),g ₁(x ₂ ,y ₂))   (28)

then we have the following relationship between the determinants of theforward and reverse remappings:

$\begin{matrix}{1 = {\left\{ {{\frac{\partial f_{2}}{\partial x}\frac{\partial g_{2}}{\partial y}} - \left( \frac{\partial g_{2}}{\partial x} \right)^{2}} \right\} _{({x_{1},y_{1}})}{\left\{ {\frac{\partial f_{1}}{\partial x}\frac{\partial g_{1}}{\partial y}\left( \frac{\partial g_{1}}{\partial y} \right)^{2}} \right\} _{({x_{2},y_{2}})}}}} & (29)\end{matrix}$

We thus focus on the inverse problem, bottom panel of FIG. 2, thatconsists of first finding the surface of h₁ as the solution of:

$\begin{matrix}{{\left\{ {\left\lbrack {E_{{DM}\; 2}\left( {f_{2},g_{2}} \right)} \right\rbrack^{2}\left\lbrack {{\frac{\partial f_{2}}{\partial x}\frac{\partial g_{2}}{\partial y}} - \left( \frac{\partial g_{2}}{\partial x} \right)^{2}} \right\rbrack} \right\} _{({x_{1},y_{1}})}} = \left\lbrack {E_{{DM}\; 1}\left( {x_{1},y_{1}} \right)} \right\rbrack^{2}} & (30)\end{matrix}$

Since our goal is to obtain a pupil as uniform a possible we seek afield at DM2 as uniform as possible:

E _(DM2)(f ₂(x ₁ ,y ₁),g ₂(x ₁ ,y ₁))=√{square root over (∫_(a4) E_(DM1)(x,y)² dxdy)}=Constant   (31)

Moreover we are only interested in compensating asymmetric structureslocated between the secondary and the edge of the primary. We thus onlyseek to find (f₂,g₂) such that:

$\begin{matrix}\begin{matrix}{{E_{{DM}\; 1}\left( {x_{1},y_{1}} \right)} = {A\left( {x_{1},y_{1}} \right)}} \\{= {\left( {\left\lbrack {{P\left( {x,y} \right)} - \left( {1 - {P_{o}\left( {x,y} \right)}} \right)} \right\rbrack*^{- \frac{x^{2} + y^{2}}{\omega^{2}}}} \right)_{({x_{1},y_{1}})}}}\end{matrix} & (32)\end{matrix}$

where P₀(x₁,y₁) is the obscured pupil, without segments or secondarysupports. Finally, we focus on solutions with a high contrast only up toa finite OWA. We artificially taper the discontinuities by convolvingthe control term in the Monge-Ampere Equation, [P(x,y)−(1−P₀(x,y)], witha gaussian of width ω, Note that this tapering is only applied whencalculating the DM shapes via solving the reverse problem. When theresulting solutions are propagated through Eq. 26 we use the truetelescope pupil for E_(DM1)(x₁,y₁). The parameter ω has a significantimpact on the final post-coronagraphic contrast. Indeed we are hereworking with a merit function that is based on a pupil-plane residual,while ideally our cost function should be based on image-planeintensity. By convolving the control term in the reverse Monge-AmpereEquation, we low pass filter the discontinuities. This is equivalent togiving a stronger weight to low-to-mid spatial frequencies of interestin the context of exo-planets imaging. For each case presented in §. 6we calculate our DM shapes over a grid of values of ω which correspondto low pass filters with cutoff frequencies ranging from ˜OWA to ˜2 OWAand we keep the shapes which yield the best contrast. This somewhatad-hoc approach can certainly be optimized for higher contrasts. Howeversuch an optimization is beyond the scope of the present manuscript.

The problem we are seeking to solve is illustrated in the second panelof FIG. 2. In this configuration the full second order Monge-AmpereEquation can be written as:

$\begin{matrix}{{{\left( {1 + {Z\frac{\partial^{2}h_{1}}{\partial x^{2}}}} \right)\left( {1 + {Z\frac{\partial^{2}h_{1}}{\partial y^{2}}}} \right)} - \left( {Z\frac{\partial^{2}h_{1}}{{\partial x}{\partial y}}} \right)^{2}} = {A\left( {x,y} \right)}^{2}} & (33)\end{matrix}$

where we have dropped the (x₁,y₁) dependence for clarity. Since we areinterested in surface deformations which can realistically be createdusing a DM, we seek for a Fourier representation of the DMs surface:

$\begin{matrix}\begin{matrix}{{h_{1}\left( {x,y} \right)} = {\frac{D^{2}}{Z}{H_{1}\left( {X,Y} \right)}}} \\{= {\frac{D^{2}}{Z}{\sum\limits_{n = {{- N}/2}}^{N/2}{\sum\limits_{m = {{- N}/2}}^{N/2}{a_{m,n}^{\frac{2\pi}{D}{({{mX} + {nX}})}}}}}}}\end{matrix} & (34)\end{matrix}$

with

  a_(−m, −n) = a_(−m, −n)^(*), a_(−m, −n) = ?, ?indicates text missing or illegible when filed

where N is the limited number of actuators across the DM. Note that wehave normalized the dimensions in the pupil plane X=x/D, Y=y/D. Thenormalized second order Monge-Ampere Equation is then:

$\begin{matrix}{{{\left( {1 + \frac{\partial^{2}H_{1}}{\partial X^{2}}} \right)\left( {1 + \frac{\partial^{2}H_{1}}{\partial Y^{2}}} \right)} - \left( \frac{\partial^{2}H_{1}}{{\partial X}{\partial Y}} \right)^{2}} = {A\left( {X,Y} \right)}^{2}} & (35)\end{matrix}$

For each configuration in this paper we first solve Eq. 35 and thentransform the normalized solution in physical units, which depends onthe DMs diameter D and their separation Z.

4.2 Solving the Monge-Ampere Equation to Find H₁

Over the past few years a number of numerical algorithms aimed atsolving Eq. 35 have emerged in the literature [Loeper & Rapetti (2005),Benamou et˜al. (2010)]. Here we summarize our implementation of two ofthem: an explicit Newton method [Loeper &Rapetti(2005)], and asemi-implicit method [Froese & Oberman (2012)]. We do not delve into theproof of convergence of each method, they can be found in Loeper &Rapetti (2005), Benamou et˜al.(2010) Froese & Oberman(2012)]. Note thatZheligovsky et˜al. (2010) discussed both approaches in a cosmologicalcontext and devised Fourier based solutions. Here we are interested in atwo dimensional problem and we outline below the essence of eachalgorithm.

4.2.1 Explicit Newton Algorithm

This method was first introduced by [Loeper & Rapetti (2005)] and relieson the fact that Eq. 35 can be re-written as

$\begin{matrix}{{\det \left\lbrack \begin{pmatrix}{1 + \frac{\partial^{2}H_{1}}{\partial X^{2}}} & \frac{\partial^{2}H_{1}}{{\partial X}{\partial Y}} \\\frac{\partial^{2}H_{1}}{{\partial X}{\partial Y}} & {1 + \frac{\partial^{2}H_{1}}{\partial Y^{2}}}\end{pmatrix} \right\rbrack} = {\det \left\lbrack {{Id} + {\mathcal{H}\left( {H_{1}\left( {X,Y} \right)} \right)}} \right\rbrack}} & (36)\end{matrix}$

-   -   where        (·) is the two dimensional Hessian of a scalar field and Id the        identity matrix. If one writes H₁=u+v with ∥v∥<<∥u∥ then:

det[Id+

(u+δv)]=det[Id+

(u)]+δ Tr[(Id+

(u))^(t) ^(T)

(v)]+o(δ²)   (37)

-   -   where (·)^(t) ^(T) denotes the transpose of the comatrix. Eq. 35        can thus be linearized as:

$\begin{matrix}{{{\left( {1 + \frac{\partial^{2}u}{\partial Y^{2}}} \right)\frac{\partial^{2}v}{\partial X^{2}}} + {\left( {1 + \frac{\partial^{2}u}{\partial X^{2}}} \right)\frac{\partial^{2}v}{\partial Y^{2}}} - {2\frac{\partial^{2}u}{{\partial X}{\partial Y}}\frac{\partial^{2}v}{{\partial X}{\partial Y}}}} = \left( {{A\left( {X,Y} \right)}^{2} - {\det \left\lbrack {\mathcal{H}\left( {\frac{X^{2} + Y^{2}}{2} + u} \right)} \right\rbrack}} \right)} & (38)\end{matrix}$

The explicit Newton algorithm relies on Eq. 38 and can then besummarized as carrying out the following iterations:

Choose a first guess H₁ ⁰.

At each iteration k we seek for a solution of the form H₁ ^(k+1)=H₁^(k)+V^(k), where V^(k) is the DM shape update.

In order to find V^(k) we write:

$\begin{matrix}{{L_{E}\left( {H_{1}^{k},V^{k}} \right)} = {{\left( {1 + \frac{\partial^{2}H_{1}^{k}}{\partial Y^{2}}} \right)\frac{\partial^{2}V^{k}}{\partial X^{2}}} + {\left( {1 + \frac{\partial^{2}H_{1}^{k}}{\partial X^{2}}} \right)\frac{\partial^{2}V^{k}}{\partial Y^{2}}} - {2\frac{\partial^{2}H_{1}^{k}}{{\partial X}{\partial Y}}\frac{\partial^{2}V^{k}}{{\partial X}{\partial Y}}}}} & (39) \\{\mspace{79mu} {{R_{E}\left( H_{1}^{k} \right)} = {\frac{1}{r}\left( {A^{2} - {\det \left\lbrack {{Id} + {\mathcal{H}\left( H_{1}^{k} \right)}} \right\rbrack}} \right)}}} & (40)\end{matrix}$

and solve

L _(E)(H ₁ ^(k) ,V ^(k))=R _(E)(H ₁ ^(k)).   (41)

Eq. 41 is a linear partial differential equation in V^(k). Since we areinterested in a solution which can be expanded in a Fourier series wewrite V^(k) as:

$\begin{matrix}{{V_{1}^{k}\left( {X,Y} \right)} = {\sum\limits_{n = {{- N}/2}}^{N/2}\; {\sum\limits_{m = {{- N}/2}}^{N/2}{v_{m,n}^{k}{^{\frac{2\; \pi}{D}{({{mX} + {nX}})}}.}}}}} & (42)\end{matrix}$

Both the right hand side and the left hand side of Eq. 45 can be writtenas a Fourier series, with a spatial frequency content between −N and Ncycles per aperture. Equating each Fourier coefficient in these twoseries yields the following linear system of (2N+1)² equations with(N+1)² unknowns.

For all m ₀ , n ₀ ε [−N,N]: ∫ _(DM) e ^(t2π(m) ⁰ ^(X+n) ⁰ ^(Y)) [L_(E)(H ₁ ^(k) ,V ^(k))−R _(E)(H ₁ ^(k))]dX dY=0   (43)

When searching for V^(k) as a Fourier series over the square geometrychosen here, this inverse problem is always well posed.

Update the solution H₁ ^(k+1)=H₁ ^(k)+V₁ ^(k+1)

The convergence of this algorithm relies on the introduction of adamping constant τ>1. [Loeper & Rapetti (2005)] showed that as long as

$\frac{X^{2} + Y^{2}}{2} + H_{1}^{k}$

remains convex, which is always true for ACAD with reasonably smallaperture discontinuities, there exists a τ large enough so that thisalgorithm converge towards a solution of Eq. 35. However since thisalgorithm is gradient based, it is not guaranteed that it converges tothe global minimum of the underlying non-linear problem. In order toavoid having this solver stall in a local minimum we follow themethodology outlined by [Froese & Oberman (2012)] and first carry out aseries of implicit iterations to get within a reasonable neighborhood ofthe global minimum.

Virtual field at M1 when solving the reverse problem. A (X₁,Y₁) is thedesired apodization (top), A^(n)(X₁,Y₁) is the apodization obtainedafter solving the Monge-Ampere Equation (center). The bottom panel showsthe difference between the two quantities: the bulk of the energy in theresidual is located in high spatial frequencies that cannot becontrolled by the DMs.

4.2.2 Implicit Algorithm

This algorithm, along with its convergence proof, is thoroughlyexplained in [Froese Oberman(2012)]. It relies on rewriting Eq. 35 as:

$\begin{matrix}{{\frac{\partial^{2}H_{1}}{\partial X^{2}} + \frac{\partial^{2}H_{1}}{\partial Y^{2}}} = \sqrt{{\det \left\lbrack {{Id} + {\mathcal{H}\left( {H_{1}\left( {X,Y} \right)} \right)}} \right\rbrack}^{2} + {2\; {A\left( {X,Y} \right)}^{2}}}} & (44)\end{matrix}$

The implicit method consists of carrying out the following iterations:

Choose a first guess H₁ ⁰

In order to find H^(k+1) we write:

${L_{I}\left( {H_{1}^{k},V^{k}} \right)} = {\frac{\partial^{2}H_{1}^{k + 1}}{\partial X^{2}} + \frac{\partial^{2}H_{1}^{k + 1}}{\partial Y^{2}}}$${R_{E}\left( H_{1}^{k} \right)} = \sqrt{{\det \left\lbrack {{Id} + {\mathcal{H}\left( {H_{1}\left( {X,Y} \right)} \right)}} \right\rbrack}^{2} + {2\; {A\left( {X,Y} \right)}^{2}}}$

and solve

L ₁(H ₁ ^(k+1))=R ₁(H ₁ ^(k)).   (45)

This problem is a linear system of (N+1)² equations with (N+1)² unknownsand can be solved using projections on a Fourier Basis:

For all m ₀ , n ₀ ε [−N,N]: ∫ _(DM) e ^(t2π(m) ⁰ ^(X+n) ⁰ ^(Y)) [L ₁(H ₁^(k) ,V ^(k))−R ₁(H ₁ ^(k))]dX dY=0   (46)

Note that the term under the square root in R₁(H₁ ^(k)) is guaranteed tobe positive at each iteration.

Iterate over k

The inverse problem in Eq. 46 is always well posed, for any basis set orpupil geometry, while the explicit Newton method runs into convergenceissues when not using a Fourier basis over a square. When seeking to usea basis set that is more adapted to the geometry of the spiders andsegments or when using a trial influence function basis for the DM, theimplicit method is the most promising method. In this paper we havelimited our scope to solving the reverse problem in the bottom panel ofFIG. 4, and used a Fourier representation for the DM, we are able to useboth methods. In order to make sure that the algorithm converges towardsthe true solution of Eq. 35 we first run a few tens of iterations of theimplicit method and, once it has converged, we seek for a more accuratesolutions using the Newton algorithm. Typical results are shown in FIG.6 where most of the residual error resides in the high spatial frequencycontent (e.g. above N cycles per aperture). Our solutions are limited bythe non-optimality of the Fourier basis to describe the mostly radialand azimuthal structures present in telescopes' apertures. Moreover theDM shape is the result of the minimization of a least squares residualin the virtual end-plane of the reverse problem, with little regard tothe spatial frequency content of the solution in the final image planeof the coronagraph. While this method yields significant contrastimprovements, as reported in §. 6, we discuss in §. 7 how it can berefined for higher contrast.

4.3 Deformation of the Second Mirror

Once the surface of DM1 has been calculated as a solution of Eq. 26, wecompute the surface of DM2 based on Eqs. 19, which stem from enforcingflatness of the outgoing on-axis wavefront. We seek a Fourierrepresentation for the surface of DM2:

$\begin{matrix}\begin{matrix}{{h_{2}\left( {x,y} \right)} = {\frac{D^{2}}{Z}{H_{2}\left( {X,Y} \right)}}} \\{= {\frac{D^{2}}{Z}{\sum\limits_{n = {{- N}/2}}^{N/2}\; {\sum\limits_{m = {{- N}/2}}^{N/2}{b_{m,n}^{\frac{2\; \pi}{D}{({{mX} + {nX}})}}}}}}}\end{matrix} & (47)\end{matrix}$

Plugging the solution found in the previous step for h₁ into Eqs. 17yields a closed form for the normalized remapping functions, (F₂,G₂):

F ₂(X ₁ ,Y ₁)=X ₁−Σ_(n=−N/2) ^(N/2) Σ_(m=−N/2) ^(N/2) i2πm a _(m,n) e^(i2π(mX) ¹ ^(+nY) ¹ ⁾

G ₂(X ₁ ,Y ₁)=Y ₁−Σ_(n=−N/2) ^(N/2) Σ_(m=−N/2) ^(N/2) i2πn a _(m,n)e^(i2π(mX) ² ^(+nY) ² ⁾

Then the normalized version of Eqs. 19 can be rewritten as:

L _(x)(X ₁ ,Y ₁)=Σ_(m=−N/2) ^(N/2) i2πm b _(m,n) e ^(i2π(mF) ² ^((X) ¹^(,Y) ¹ ^()+nF) ² ^((X) ¹ ^(,Y) ¹ ⁾⁾

R _(x)(X ₁ ,Y ₁)=X ₁ −F ₂(X ₁ ,Y ₁)

L_(x)=R_(x)   (48)

L _(y)(X ₁ ,Y ₁)=Σ_(m=−N/2) ^(N/2) i2πn b _(m,n) e ^(i2π(mF) ² ^((X) ¹^(,Y) ¹ ^()+nF) ² ^((X) ¹ ,Y ¹ ⁾⁾

R _(y)(X ₁ ,Y ₁)=Y ₁ −G ₂(X ₁ ,Y ₁)

L_(y)=R_(y)   (49)

We then multiply each side of Eq. 48 and Eq. 49 by:

e ^(i2π(m) ⁰ ^(F) ^(2(X1) ^(,Y) ¹ ^()+n) ⁰ ^(F) ² ^((X) ¹ ^(,Y) ₁ ⁾⁾det[Id+

(H ₁ ^(k)(X ₁ ,Y ₁))]

where (m₀,n₀) corresponds to a given DM spatial frequency. Integratingover the square area of the DM and using the orthogonality of theFourier basis yields the following system of 2*(N+1)² equations with(N+1)² real unknowns:

  For  all (m₀, n₀):2 π   m₀b_(m₀, n₀) = ∫_(DM) R_(x)(X, Y)det [Id + ℋ(H₁^(k)(X, Y))]^( 2 π(m₀F₂(X, Y) + n₀F₂(X, Y)))XY  For  all (m₀, n₀):2 π   n₀b_(m₀, n₀) = ∫_(DM) R_(x)(X, Y)det [Id + ℋ(H₁^(k)(X, Y))]^( 2 π(m₀F₂(X, Y) + n₀F₂(X, Y)))XY2 π  n₀b_(m₀, n₀) = ∫_(DM) R_(y)(X, Y)det [Id + ℋ(H₁^(k)(X, Y))]^( 2 π(m₀F₂(X, Y) + n₀F₂(X, Y)))XY

We then find H₂, the normalized surface of DM2, by solving this systemin the least squares sense. Once the Monge-Ampere Equation has beensolved, the calculation of the surface of the second mirror is a mucheasier problem. Indeed, by virtue of the conservation of the on-axisoptical path length, finding the surface of DM2 only consists of solvinga linear system (see [Traub & Vanderbei (2003)]).

4.4 Boundary Conditions

The method described above does not enforce any boundary conditionsassociated with Eq. 35. One practical set of boundary conditionsconsists of forcing the edges of each DM to map to each other:

$\begin{matrix}{{F_{i}\left( {{\pm \frac{1}{2}},Y} \right)} = {\pm \frac{1}{2}}} & (50) \\{{F_{i}\left( {X,{\pm \frac{1}{2}}} \right)} = X} & (51) \\{{G_{i}\left( {{\pm \frac{1}{2}},Y} \right)} = Y} & (52) \\{{G_{i}\left( {X,{\pm \frac{1}{2}}} \right)} = {\pm \frac{1}{2}}} & (53)\end{matrix}$

with i=1,2. These correspond to a set of Neumann boundary conditions inH₁(X,Y) and H₂(X,Y). These boundary conditions can be enforced byaugmenting the dimensionality of the linear systems on Eq. 43 and Eq.46, however doing so increases the residual least squares errors andthus hampers the contrast of the final solution. Moreover FIG. 4 andFIG. 5 show that, because of the one to one remapping near the DM edgesin the control term of the reverse problem, the boundary conditions arealmost met in practice. For the remainder of this paper we thus do notinclude boundary conditions when calculating the DM shapes, when solvingfor H₁(X,Y) in Eq. 35 since, in the worse case, only the edge rows andcolumns of the DMs actuator will have to be sacrificed in order for theedges to truly map to each other.

4.5 Remapped Aperture

For a given pupil geometry we have calculated (H₁,H₂). We then convertthe DM surfaces to real units, (h₁,h₂), by multiplication with D²/Z. Weevaluate the remapping functions using Eqs. 17 and 19 and obtain thefield at the entrance of the coronagraph in the ray optic approximation

$\begin{matrix}{{E_{{DM}\; 2}\left( {x_{2},y_{2}} \right)} = {\left\{ {\sqrt{\det \lbrack J\rbrack}{E_{{DM}\; 1}\left( {f_{1},g_{1}} \right)}^{\frac{2\; \pi}{\lambda}{({{S{({f_{1},g_{1}})}} + {h_{1}{({f_{1},g_{1}})}} - h_{2}})}}} \right\} _{({x_{2},y_{2}})}}} & (54)\end{matrix}$

where the exponential factor corresponds to the Optical Path Lengththrough the two DMs. Even if the surface of the DMs has been calculatedusing only a finite set of Fourier modes, we check that the optical pathlength is conserved.

FIG. 6 shows that since the curvature of the DMs is limited by thenumber of modes N, our solution does not fully map out thediscontinuities induced by the secondary supports and the segments.However, they are significantly thinner and one can expect that theirimpact on contrast will be attenuated by orders of magnitude. In orderto quantify the final coronagraphic contrasts of our solution we thenpropagate it through an APLC coronagraph designed using the method in §.2. In the case of a hexagon based primary (such as JWST), we use acoronographic apodizer with a slightly oversized secondary obscurationand undersize outer edge in order to circularize the pupil. Note thatthis choice is mainly driven by the type of coronagraph we chose in §. 2to illustrate our technique. Since the DM control strategy presented inthis section is independent of the coronagraph, it can be generalized toany of the starlight suppression systems which have been discussed inthe literature. For succinctness we present our results usingcoronagraph solely based on using pupil apodization (either in an APLCor in a PIAAC configuration). Results for a JWST geometry are shown onFIG. 5.3 and FIG. 7 and discussed in §. 6. Eq. 54 assumes that thepropagation between the two DMs occurs according to the laws of rayoptics. In the next section we derive the actual diffractive field atDM2, e.g Eq. 23, and show that in the pupil remapping regime of ACAD,edge ringing due to the free space propagation is actually smaller thanin the Fresnel regime.

5 Chromatic Properties 5.1 Analytical Expression of the Diffracted Field

ACAD is based on ray optics. It is an inherently broadband technique,and provided that the coronagraph is optimized for broadband performanceACAD will provide high contrast over large spectral windows. However,when taking into account the edge diffraction effects that are capturedby the quadratic integral in Eq. 23, the true propagated field at DM2becomes wavelength dependent. More specifically, when λ is not zero thenthe oscillatory integral superposes on the ray optics field a series ofhigh spatial frequency oscillations. In theory, it would be best to usethis as the full transfer function to include chromatic effects in thecomputation of the DMs shapes. However, as discussed in §. 4, solvingthe non-linear Monge-Ampere Equation is already a delicate exercise, andwe thus have limited the scope of this paper to ray optics solutions.Nonetheless, once the DMs' shapes have been determined using ray optics,one should check whether or not the oscillations due to edge diffractionwill hamper the contrast. This approach is reminiscent of the design ofPIAA systems where the mirror shapes are calculated first usinggeometric optics and are then propagated through the diffractiveintegral in order to check a posteriori whether or not the chromaticdiffractive artifacts are below the design contrast [Pluzhniket˜al.(2005)]. In this section we detail the derivation of Eq. 23 thatis the diffractive integral for the two DMs remapping system and usethis formulation to discuss the diffractive properties of ACAD.

We start with the expression of the second order diffractive field atDM2 as derived in [Pueyo et˜al. (2011)], Eq. 20.

${E_{{DM}\; 2}\left( {x_{2},y_{2}} \right)} = {{\frac{1}{\; \lambda \; Z}\left\{ {\int{{E_{{DM}\; 1}\left( {x,y} \right)}^{\frac{\; \pi}{\lambda \; Z}{\lbrack{{\frac{\partial f_{2}}{\partial x}{({x - x_{1}})}^{2}} + {2\frac{\partial g_{2}}{\partial x}{({x - x_{1}})}{({y - y_{1}})}} + {\frac{\partial g_{2}}{\partial y}{({y - y_{1}})}^{2}}}\rbrack}}}} \right\}}_{({x_{1},y_{1}})}}$

We write E_(DM1)(x,y) as its inverse Fourier transform

E _(DM1)(x,y)=∫{circumflex over (E)}_(DM1)(ξ,η)e ^(i2π(xξ+yη))dξdη  (55)

and insert this expression in Eq. 20. Completing the squares in thequadratic exponential factor then yields:

$\begin{matrix}{{E_{{DM}\; 2}\left( {x_{2},y_{2}} \right)} = {{\frac{1}{\; \lambda \; Z}\left\{ {\int{{{\hat{E}}_{{DM}\; 1}\left( {\xi,\eta} \right)}{I_{1}\left( {f_{1},g_{1}} \right)}^{{2}\; {\pi {\lbrack{{f_{1}\xi} + {g_{1}\eta}}\rbrack}}}^{{- }\; \pi \; \lambda \; {Z{({\frac{\partial f_{2}}{\partial y}{_{({f_{1},g_{1}})}{\xi^{2} + \frac{\partial g_{2}}{\partial x}}}_{({f_{1},g_{1}})}\eta^{2}})}}}{\xi}{\eta}}} \right\}}_{({x_{2},y_{2}})}\mspace{20mu} {with}}} & (56) \\{{I_{1}\left( {y_{1},x_{1}} \right)} = {\left\{ {\int{{^{\frac{\; \pi}{\lambda \;}}}^{\lbrack\begin{matrix}{{\frac{\partial f_{2}}{\partial x}{({x - x_{1} - \frac{\xi \; \lambda \; Z}{\frac{\partial f_{2}}{\partial x}}})}^{2}} +} \\{{2\frac{\partial g_{2}}{\partial x}{({x - x_{1}})}{({y - y_{1}})}} + {\frac{\partial g_{2}}{\partial y}{({y - y_{1} - \frac{\eta \; \lambda \; Z}{\frac{\partial g_{2}}{\partial x}}})}^{2}}}\end{matrix}\rbrack}{x}{y}}} \right\} _{({x_{1},y_{1}})}.}} & (57)\end{matrix}$

The integral over space, I₁(y₁,x₁), is the integral of a complexgaussian and can be evaluated analytically. This yields:

$\begin{matrix}{{E_{{DM}\; 2}\left( {x_{2},y_{2}} \right)} = {\left\{ {\frac{1}{\sqrt{{\frac{{\partial f_{2}}{\partial g_{2}}}{{\partial x}{\partial y}} - \left( \frac{\partial g_{2}}{\partial x} \right)^{2}}}}{\int{{{\hat{E}}_{{DM}\; 1}\left( {\xi,\eta} \right)}^{\; 2\; {\pi {\lbrack{{x_{1}\xi} + {y_{1}\eta}}\rbrack}}}^{{- }\; \pi \; \lambda \; {Z{({{\frac{\partial f_{2}}{\partial y}\xi^{2}} + {\frac{\partial g_{2}}{\partial x}\eta^{2}}})}}}{\xi}{\eta}}}} \right\} _{({x_{1},y_{1}})}.}} & (58)\end{matrix}$

We thus have expressed E_(DM2)(x₂,y₂) as a function of(x₁,y₁)=(f₁(x₂,y₂),g₁(x₂,y₂)). This expression can be furthersimplified: using Eqs. 27 to 29 one can derive

${\frac{\partial f_{2}}{\partial x}_{({x_{1},y_{1}})}} = {{\frac{1}{\det \left\lbrack {J\left( {x_{2},y_{2}} \right)} \right\rbrack}\frac{\partial g_{1}}{\partial y}}_{({x_{2},y_{2}})}{and}}$${\frac{\partial g_{2}}{\partial x}_{({x_{1},y_{1}})}} = {{\frac{1}{\det \left\lbrack {J\left( {x_{2},y_{2}} \right)} \right\rbrack}\frac{\partial f_{1}}{\partial x}}_{({x_{2},y_{2}})}.}$

Which finishes to prove Eq. 23:

$\mspace{20mu} {{E_{{DM}\; 2}\left( {x_{2},y_{2}} \right)} = {\left\{ {\sqrt{{\det \lbrack J\rbrack}}\text{?}\left( {\xi,\eta} \right)\text{?}\text{?}{\xi}{\eta}} \right\} _{({x_{2},y_{2}})}{\text{?}\text{indicates text missing or illegible when filed}}}}$

This expression is very similar to a modified Fresnel propagation andcan be rewritten as such:

$\begin{matrix}{{E_{{DM}\; 2}\left( {x_{2},y_{2}} \right)} = {{\left\{ {\int_{}^{\;}{{E_{{DM}\; 1}\left( {x,y} \right)}^{{- }\frac{\pi \; {\det {\lbrack J\rbrack}}}{\lambda \; Z}{({{{(\frac{\partial g_{1}}{\partial y})}^{- 1}{({x - f_{1}})}^{2}} + {{(\frac{\partial f_{1}}{\partial x})}^{- 2}{({y - g_{1}})}^{2}}})}}\ {x}{y}}} \right\} _{({x_{2},y_{2}})}{E_{{DM}\; 2}\left( {x_{2},y_{2}} \right)}} = {\left\{ {\int_{}^{\;}{{E_{{DM}\; 1}\left( {x,y} \right)}^{{- }\frac{\pi \; {\det {\lbrack J\rbrack}}}{\lambda \; Z}{({{{(\frac{\partial g_{1}}{\partial y})}^{- 1}{({x - f_{1}})}^{2}} + {{(\frac{\partial f_{1}}{\partial x})}^{- 1}{({y - g_{1}})}^{2}}})}}\ {x}{y}}} \right\} _{({x_{2},y_{2}})}}}} & (60)\end{matrix}$

Because of this similarity we call this integral the Stretched-RemappedFresnel approximation (SR-Fresnel). Indeed in this approximation thepropagation distance is stretched by

$\left( {\frac{\frac{\partial g_{1}}{\partial y}}{\det \lbrack J\rbrack},\frac{\frac{\partial f_{1}}{\partial x}}{\det \lbrack J\rbrack}} \right)$

and the integral is centered around the remapped pupil (f₁,g₁).

5.2 Discussion

FIG. 10, Left: Comparison of edge diffraction between PIAA and Fresnelpropagation. Because the discontinuities in the pupil occur at thelocation where Γ_(x)<1, a PIAA amplifies the chromatic ringing whencompared to a more classical propagation. Right: Comparison of edgediffraction between ACAD and Fresnel propagation. Because thediscontinuities in the pupil occur at the location where Γ_(x)>1, a ACADdamps the chromatic ringing when compared to a more classicalpropagation. These simulations were carried out for a one dimensionalaperture. In this case the Monge-Ampere Equation can be solved usingfinite elements and the calculation of the diffractive effects reducesto the evaluation of various Fresnel special functions at varyingwavelength. The full two-dimensional problem is not separable andrequires the development of novel numerical tools.

The integral form provides physical insight about the behavior of thechromatic edge oscillations. If we write

$\begin{matrix}{\Gamma_{x} = {\frac{\det \lbrack J\rbrack}{\frac{\partial g_{1}}{\partial y}}_{({x_{2},y_{2}})}}} & (61) \\{\Gamma_{y} = {\frac{\det \lbrack J\rbrack}{\frac{\partial f_{1}}{\partial x}}_{({x_{2},y_{2}})}}} & (62)\end{matrix}$

we can identify a several diffractive regimes;

-   -   1. when Γ_(x)=Γ_(y)=1 the E_(DM2)(x₂,y₂) reduces to a simple        Fresnel propagation. Edge ringing can them be mitigated using        classical techniques such as pre-apodization, or re-imaging into        a conjugate plane using oversized relay optics.    -   2. when Γ_(x),Γ_(y)<1 then it is as if the effective propagation        length through the remapping unit was increased. This magnifies        the edge chromatic ringing.    -   3. when Γ_(x),Γ_(y)>1 then it is as if the effective propagation        length through the remapping unit was decreased. This damps the        edge chromatic ringing.    -   4. when Γ_(x)>1 and Γ_(y)<1, e.g. at a saddle point in the DM        surface, then it is as if the effective propagation length        through the remapping unit was decreased in one direction and        increased in the other. The edge chromatic ringing can either be        damped or magnified depending on the relative magnitude of Γ_(x)        and Γ_(y).

In the case of a PIAA coronagraphs, the mirror shapes are such thatΓ_(x), Γ_(y)>1 at the center of the pupil and Γ_(x),Γ_(y)<<1 at theedges of the pupil, where the discontinuities occur. As a consequencethe edge oscillations are largely magnified when compared to Fresneloscillations (see right panel of FIG. 10), and apodizing screens arenecessary in order to reduce the local curvature of the mirror's shape(as also discussed in [Pluzhnik et˜al. (2005), Pueyo et˜al, (2011)]. Inthe case of ACAD, where the x-axis is chosen to be perpendicular to thediscontinuity, the surface curvature is such that Γ_(x)>1, Γ_(y)˜1 atthe discontinuities inside the pupil and Γ_(x)>1, Γ_(y)˜1 elsewhere.This yields damped chromatic oscillations at the remappeddiscontinuities and Fresnel oscillations at the edges of the pupil (seeright panel of FIG. 10). Note that FIG. 10 was generated using a onedimensional toy model that assumes Eq. 23 is separable, e.g Γ_(y)=1, asdescribed in [Pueyo et˜al.(2011)]. In practice at the saddle points ofthe optical surfaces, near the junction of two spiders for instance,γ_(x)>1, ⊖_(y)<˜1 and thus our separable model does not guarantee thanin the true 2D case chromatic edge oscillations might not be locallyamplified. However even near the saddle points ACAD provides a stronglyconverging remapping in the direction perpendicular to the discontinuityand very little diverging remapping in the other direction. As aconsequence Γ_(x)>>1 and Γ_(y) is smaller than but close to one. We thuspredict that even at these locations chromatic ringing will not beamplified. Even though ACAD is based on pupil remapping, its diffractionproperties are qualitatively very different from PIAA coronagraphs sinceedge ringing is not amplified beyond the Fresnel regime at the pupiledges, and is attenuated near the discontinuities. We conclude that inmost cases ACAD operates in a regime where edge chromatic oscillationsare not larger than classical Fresnel oscillations, and sometimesactually smaller. As a consequence the chromaticity of this ringing canbe mitigated using standard techniques developed in the Fresnel regime,and we do not expect this phenomenon to be a major obstacle to ACADbroadband operations.

5.3 Diffraction Artifacts in ACAD are no Worse than Fresnel Ringing

Results obtained when applying our approach to a geometry similar toJWST. We used two 3 cm DMs of 64 actuators separated by 1 m. Theirmaximal surface deformation is 1.1 μm, well within the stroke limit ofcurrent DM technologies. The residual light in the corrected PSF followsthe secondary support structures and can potentially be furthercancelled by controlling the DMs using an image plane based costfunction, see FIG. 23

We have established that the diffractive chromatic oscillationsintroduced by the fact that DM1 and DM2 are not located in conjugateplanes is no worse than classical Fresnel ringing from the apertureedges and can be mitigated using well-know techniques which have beendeveloped for this regime. While a quantitive tradeoff study of how todesign a high contrast instrument which minimizes such oscillations isbeyond the scope of this paper, we briefly remind their qualitativeessence to the reader:

The edges of the discontinuities in the telescope aperture can besmoothed via pupil apodization before DM1. This solution is notparticularly appealing as it requires the introduction of atransmissive, and thus dispersive, component in the optical train.

The distance between the two DMs can be reduced. Indeed the DMsdeformations presented herein, for 3 cm DMs separated by 1 m, are all ≦1μm while current technologies allow for deformations of several microns.As the edge ringing scales as Z/D² chromatic oscillations will bereduced by decreasing Z. Since the DM surfaces scale as D²/Z reducing Zwill increase the DM deformations but have little impact on thefeasibility of our solution as current DM technologies can reach 4 μmstrokes.

The coronagrahic apodizer can be placed in a plane that is conjugate tothe DM1. This can be achieved by re-imaging DM2 through a system ofoversized optics (the over-sizing factor increases steeply when thepupil diameter decreases). By definition there are no Fresnel edgeoscillations in such a plane. Alternatively a coronagraph without apupil apodization (amplitude or phase mask in the image plane) can beused, and in this configuration it is only sufficient for the optics tobe oversized.

Note that these three solutions are not mutually exclusive and that onlya full diffractive analysis, which uses robust numerical propagatorsthat have been developed based on Eq. 23, can quantitatively address thetrade-offs discussed above. The development of such propagators is ournext priority. In [Pueyo et˜al. (2011)] we laid out the theoreticalfoundations of such a numerical tool in the case of circularly symmetricpupil remapping and this solution has been since then practicallyimplemented, as reported by [Krist et˜al, (2010)]. Generalizing thismethod to a tractable propagator in the case of arbitrary remapping is ayet unsolved computational problem. In the meantime we emphasize thatwhile the spectral bandwidth of coronagraphs whose incoming amplitudehas been corrected using ACAD will certainly be limited by edgediffraction effects, but these effects are no worse than Fresnel ringingand can thus be mitigated using optical designs which are now routinelyused in high contrast instruments (see [Vàinaud et˜al. (2010)] for suchdiscussions). For the remainder of this paper we thus assume thediffractive artifacts have been adequately mitigated and we compute ourresults assuming a geometric propagation between DM1 and DM2.

When using two sequential DMs as a pure amplitude actuator (e.g.,assuming a flat wavefront after the second DM) the relationship betweenthe surface deformations and the outgoing electrical field is given bythe Huygens integral. We derived, under the second order approximationfor the Huygens integrand, a simplified expression of this diffractivetransfer function:

$E_{{DM}\; s} = {\left\{ {\frac{1}{\sqrt{{{\frac{\partial f_{2}}{\partial x}\frac{\partial g_{2}}{\partial y}} - \left( \frac{\partial g_{2}}{\partial x} \right)^{2}}}}{\int{{E_{{DM}\; 1}\left( {\xi,\eta} \right)}^{\; 2\; {\pi {\lbrack{{x_{1}\xi} + {y_{1}\eta}}\rbrack}}}^{{- }\; \pi \; \lambda \; {Z{\lbrack{{\frac{\partial f_{2}}{\partial y}\xi^{2}} + {\frac{\partial g_{2}}{\partial x}\eta^{2}}}\rbrack}}}{\xi}{\eta}}}} \right\} _{({x_{2},y_{2}})}}$

The computational cost associated with the diffractive transfer functionin the above equation and also Eq. (59) makes it almost impossible todevise a global inversion method that captures both the remapping oflarge amplitude errors and the diffractive properties of the dual DMsunit.

Procedure to Include Diffraction

As a consequence in order to find DM shapes that efficiently compensatefor the secondary support structures, we provide the procedure outlinedbelow:

-   -   1. First invert the problem in the ray optics approximation in        order to map out the pupil imprint of the spiders as much as        possible, as discussed in this specification.    -   2. Second compute in the SR-approximation the diffractive        response associated with the derived DM shapes using Eq. (59).        Assess the broadband properties of the dual DM unit in the        presence of diffractive effects.    -   3. Using a small perturbation superposed to the deformations        calculated in step 1, compute, around the equilibrium point set        by step 1, the sensitivity matrix for both DMs seen through the        chosen coronagraph. This matrix is calculated using the        diffractive models in step 2, e.g. in the SR-approximations. We        conduct an iterative closed loop.

Results for the Astronomical Telescope Application

As an illustration we have applied the entire methodology describedabove to a geometry analogous to the AFTA telescopes. We have inparticular successfully implemented a numerical integrator capable ofcapturing the diffractive effects in Eq.(59) that are the sources of thebandwidth limit in ACAD. Our results are shown on FIG. 26 where wecalculated, assuming that the DMs have been controlled according to step1, the field in a plane conjugate with the telescope pupil. Whenintroducing a Focal Plane Mask, this plane would be the Lyot plane ofthe coronagraph and its wavefront topology drives the contrast floor andbandwidth when using linear DM control loops. From a ray opticsperspective ACAD “fills up as much as possible” the area obscured by thesecondary support structure FIG. 3, while in the diffractive regime wecan see that ACAD replaces the deep and sharp amplitude errors fromsupports with shallow and smooth ripples. The chromatic nature of theseoscillations sets the ultimate bandwidth of ACAD. We see that ACAD onlyweakly impacts the bandwidth of the starlight suppression system, asshown in FIG. 27 where we have combined these diffractive models with aRadial Apodization Vector Vortex Coronagraph (RA-VVC, Mawet et al., inprep., and obtained contrasts as deep at 3×10at $2\lambda/D$ over thefull range of wavelengths from 500 nm to 700 nm, a bandwidth of over30%. The combination of FIG. 23 (APLC or PIAAC) and FIG. 27 illustratesthe second advantage of ACAD: it is fundamentally “coronagraphagnostic”. We have conducted diffractive simulations using the RA-VVCsince it is one of the most promising technologies for obscuredapertures 70% throughput in its high transmissivity configuration andtheoretical IWA as close as 1.5 λ/D. Smaller IWA solutions should theybe available later on.

6 Results 6.1 Application to Future Observatories 6.1.1 JWST

We have illustrated each step of the calculation of the DM shapes in §.4 using a geometry similar of JWST. This configuration is somewhat aconservative illustration of an on-axis segmented telescope as itfeatures thick secondary supports and a “small” number of segments whosegaps diffract light in regions of the image plane close to the opticalaxis (the first diffraction order of a six hexagons structure is locatedat ˜3λ/D). In order to assess the performances of ACAD on such anobservatory architecture we chose to use a coronagraph designed around aslightly oversized secondary obscuration of diameter 0.25 D, with afocal plane mask of diameter 3λ/D, an IWA of 5λ/D and an OWA of 30λ/D.The field at the entrance of the coronagraph after remapping by the DMsis shown on the top right panel of FIG. 11. The DM surfaces, calculatedassuming 64 actuators across the pupil (N=64 in the Fourier expansion)and DMs of diameter 3 cm separated by Z=1 m, are shown on the middlepanel of FIG. 11. They are well within the stroke limit of current DMtechnologies. The surfaces were calculated by solving the reverseproblem over an even grid of 10 cutoff low-pass spatial frequenciesranging between 30 and 70 cycles per apertures for the tapering kernelω. The value yielding the best contrast was chosen. Note that theoptimal cutoff frequency depends on the spatial scale of thediscontinuities, and that higher contrasts could be obtained by choosinga set of two convolution kernels for the reverse problem and finding theoptimal solution using a finer grid. However, the results in the bottomrow of FIG. 11 are extremely promising. FIG. 12 shows a contrastimprovement of a factor of 100 when compared to the raw coronagraphicPSF, which is quite remarkable for an algorithm which is not based on animage-plane metric. These results illustrate that even with a veryunfriendly aperture similar to JWST one can obtain contrasts as high asenvisioned for upcoming Ex-AO instruments, which have been designed formuch friendlier apertures. While we certainly do not advocate to usesuch a technique on JWST, this demonstrates that ACAD is a powerful toolfor coronagraphy with on-axis segmented apertures.

6.1.2 Extremely Large Telescopes

PSFs resulting from ACAD when varying the number and thickness ofsecondary support structures while maintaining their covered surfaceconstant (FIG. 15). The surface area covered in this example is 5096greater than in the TMT example shown on FIG. 13. As the spiders getthinner their impact on raw contrast becomes smaller and the starlightsuppression after DM correction becomes bigger. For a relatively smallnumber of spiders (<12) the contrast improvement on each singlestructure is the dominant phenomenon, regardless of the number ofspiders. ELTs designed with a moderate to large number of thin secondarysupport structure (6 to 12) present aperture discontinuities which areeasy to correct with ACAD.

We now discuss the case of Extremely Large Telescopes and provide anillustration using the example of the Thirty Meter Telescope. Weconsidered the aperture geometry shown on the top left panel of FIG. 13.It consists of a pupil 37 segments across in the longest direction and asecondary of diameter ˜0.12D which is held by three main thick strutsand six thin cables. As seen on the bottom left panel of FIG. 13 theimpact of segment gaps is minor as they diffract light beyond the OWA ofthe coronagraph. When using a coronagraph with a larger OWA the segmentgaps will have to be taken into account, and will have to be mitigatedusing DMs with a larger number of actuators. In order to obtain firstorder estimates of the performances of ACAD on the aperture geometryshown on the top left panel of FIG. 13, we chose to use a coronagraphdesigned around a slightly oversized secondary obscuration of diameter0.15 D, with a focal plane mask of 6λ/D diameter, an IWA of 4λ/D and anOWA of 30λ/D. The field at the entrance of the coronagraph afterremapping by the DMs is shown on the top right panel of FIG. 13. The DMsurfaces, calculated assuming 64 actuators across the pupil (N=64 in theFourier expansion) and DMs of diameter 3 cm separated by Z=1 m, areshown on the middle panel of FIG. 13. They are well within the strokelimit of current DM technologies. The surfaces were calculated bysolving the reverse problem over an even grid of 10 cutoff low-passspatial frequencies ranging between 30 and 70 cycles per apertures forthe tapering kernel ω. The value yielding the best contrast was chosen.The final PSF is shown on the bottom right panel of FIG. 13 and featuresa high contrast dark hole with residual diffracted light at the locationof the spiders' diffraction structures. The impact on coronagraphiccontrast of secondary supports was thoroughly studied by [Martinezet˜al. (2008)]. They concluded that under a 90% Strehl ratio, thecontrast in most types of coronagraphs is driven by the secondarysupport structures to levels ranging from 10⁻⁴ to 10⁻⁵. This, in turn,leads to a final contrast after post-processing (called DifferentialImaging) of ˜10⁻⁷-10⁻⁸. FIG. 14 shows that using ACAD on an ELT pupilyields contrasts before any post-processing which are comparable to theones obtained by [Martinez et˜al. (2008)] after Differential Imaging.This demonstrates that should two sequential DMs be integrated into afuture planet finding instrument, setting their surface deformationaccording to the methodology presenting above would allow thisinstrument to perform its scientific program at a very high contrast.Moreover the surface of the DMs could be adjusted to mitigate for theeffect of missing segments at the surface of the primary (when forinstance the telescope is operating while some segments are beingserviced).

6.2 Hypothetical Cases 6.2.1 Constant Area Covered by the SecondarySupport Structures

In the case of ELTs with large number of small segments (when comparedto the aperture size), gaps diffract light far from the optical axis(see FIG. 13 for an example). The secondary support structures are thenthe major source of unfriendly coronagraphic diffracted light. Under theassumption that thick structures are necessary to support the heavysecondary over the very large ELT pupils, one can use the aperture areacovered by the spiders as a proxy of the secondary lift constraint. Wehave thus explored a series of geometries for which the number ofspiders increases as they get thinner while the overall area covered bythe secondary support structures remains constant. In the examples shownfrom FIG. 15 to FIG. 18, the area covered by the secondary supportstructures is 1.5 times greater than in the TMT geometry discussedabove. In all cases we used a coronagraph with a central obscuration of0.15 D, with a focal plane mask of 6λ/D diameter, and IWA of 4λ/D and anOWA of 30λ/D. The surfaces were calculated by solving the reverseproblem over an even grid of 10 cutoff low-pass spatial frequenciesranging between 30 and 70 cycles per apertures for the tapering kernel.The value yielding the best contrast was chosen. This exercise leads toseveral conclusions pertaining to the performances of ACAD with variouspotential ELT geometries.

Clocking of the Spiders with Respect to the DM

The top two panels of FIG. 15 illustrate the importance of the clockingof the spiders with respect to the DMs actuator grid (or the Fouriergrid in our case). When the secondary support structures are clocked by45° with respect to the DM actuators they are much more attenuated byACAD, thus yielding higher contrast. This is an artifact of the Fourierbasis set chosen and would be mitigated by using DMs whose actuatorplacement presents circular and azimuthal symmetries [Watanabe et˜al.(2008)].

Annulus in the PSF with a Large Number of Spiders

When the number of secondary support struts becomes very large (>20), aninteresting phenomena occurs in the raw PSF: the spiders diffract lightoutside an annulus of radius N_(Spiders)/πλ/D, just as spiderweb masksdo in the case of shaped pupil coronagraphs [Vanderbei et˜al. (2003)].The “bump” located beyond that spatial frequency is more difficult toattenuate using the DMs (see FIG. 15 for an illustration). ACAD createssmall ripples at the edges of the remapped discontinuities and when toomany discontinuities are in the vicinity of each other, then theseripples interfere constructively and hamper the starlight extinctionlevel yielded by ACAD.

A Lot of Thin Spiders is More Favorable Than a Few Thick Spiders

In general decreasing the width of the spiders while increasing theirnumber is beneficial to the contrast obtained after ACAD as illustratedon the radial averages on FIG. 16 and FIG. 18. When one increases thenumber of spiders while decreasing their width in a classicalcoronagraph, the peak intensity of the diffraction pattern of one spiderdecreases as the squared width of the spider. The radially averagedcontrast improvement without ACAD is then somewhat lesser than thesquare of the spider thinning factor as it is mitigated by theincreasing number of spiders. When using ACAD the spiders are seen bythe coronagraph as much thinner than they actually are (by a factor τ)and thus the peak intensity of their diffraction pattern is lower by afactor of τ², Our numerical experiments show that τ increases when thespider width decreases. As consequence, the overall contrast gain afterACAD when decreasing the width of the spiders while increasing theirnumber is greater than in the case of a classical coronagraph. Whendesigning ELT secondary support structures and planning to correct forthem using ACAD, increasing the number of spiders to 8 or even 12 has abeneficial impact on contrast as it enables each discontinuity to becomethinner and thus to be corrected to higher contrast using the DMs. ThePSFs of apertures with more than 12 spiders present diffractionstructures which are poorly suited for correction with square DMs. Whilethe contrast resulting from applying ACAD to such apertures is still adecreasing function of the number of spiders, FIG. 18 shows that the netcontrast gain brought by the DM based remapping is smaller than in themore gentle cases of 12 spiders. The study presented on FIG. 15 to FIG.18 remains to be fully optimized for each potential design of an ELTplanet finding instrument (in particular using a finer grid of cutoffspatial frequencies). It however demonstrates the flexibility of ACADfor various aperture geometries and provides a first order rule of thumbto design telescope apertures which are friendly to direct imaging ofexo-planets: “A lot of thin spiders is more favorable than a few thickspiders”. In practice the number of spiders will be limited by effectsnot treated in our analysis such as the mechanical rigidity,requirements on the perfection of their periodic spacing and glancingreflections from the sides of multiple spiders. We thus advocate that,should future ELTs be built with high contrast exo-planetary science asa main scientific driver, then such effect ought to be thoroughlyanalyzed as a large numbers of thin spiders is more favorable from acontrast standpoint when using ACAD to mitigate for pupil amplitudeasymmetries.

6.2.2 Monolithic on-Axis Apertures.

PSFs resulting from ACAD when varying the number and thickness ofsecondary support structures. As the spiders get thinner their impact onraw contrast becomes lesser and the starlight suppression after DMcorrection becomes greater. In this case ω was optimized on a very finegrid and the aperture we clocked in a favorable direction with respectto the Fourier basis.

When discussing the case of JWST we stressed the complexity associatedwith the optimization of ACAD in the presence of aperturediscontinuities of varying width. Carrying out such an exercise would beextremely valuable to study the feasibility of the direct imaging ofexo-earths with an on-axis segmented future flagship observatory such asATLAST [Postman et˜al. (2010)]. However, such an effort iscomputationally heavy and thus beyond the scope of the present paper,which focuses on introducing the ACAD methodology and illustrating usingkey basic examples.

So far, none of the examples in this manuscript demonstrate that ACADcan yield corrections all the way down to the theoretical contrast floorthat is set by the coronagraph design. When seeking to image exo-earthsfrom space, future missions will need to reach this limit. In order toexplore this regime, we conducted a detailed study of a hypotheticalon-axis monolithic telescope with four secondary support struts. Toestablish the true contrast limits we varied the thickness of thespiders for each geometry. The surfaces were calculated by solving thereverse problem over an even grid of 70 cutoff low-pass spatialfrequencies ranging between 30 and 70 cycles per apertures for thetapering kernel. The value yielding the best contrast was chosen. In allcases we used a coronagraph with a central obscuration of 0.15 D, with afocal plane mask of 6λ/D diameter, and IWA of 4λ/D, and an OWA of 30λ/D.Note that when using coronagraphs relying on pupil apodization theseresults can be readily generalized to larger circular secondaryobscurations, at a loss in IWA (as shown on FIG. 2). Moreover we clockedthe telescope aperture by 45° with respect to the grid of Fourier modes.We found that, indeed, the theoretical contrast floor set by thecoronagraph design is met for thin spiders (0.02 D) and it is very closeto being met for spiders only half the thickness of the the onescurrently equipping the Hubble Space Telecope (0.05 D), see FIG. 19 andFIG. 20. Even in the case of thick struts (0.1 D) we find contrasts anorder of magnitude higher than in the similar configuration on the toppanel of FIG. 10, due to our thorough optimization of the cutofffrequency of the tapering kernel and careful clocking of the aperturewith respect to the actuators. On-axis telescopes are thus a viableoption to image earth-analogs from space: their secondary supportstructures can be corrected down to contrast levels comparable to thetarget contrast of recent missions concept studies [Guyon et˜al. (2008),Trauger et˜al. (2010)]. Since the baseline wavefront controlarchitecture for future space coronagraphs relies on two sequential DMs,ACAD does not add any extra complexity to such missions and merelyconsists of controlling the DMs in order to optimally compensate for theeffects of asymmetric aperture discontinuities.

7 Discussion 7.1 Field Dependent Distortion

Because ACAD relies on deforming the DMs' surfaces in an asphericalfashion, off-axis wavefronts seen through the two DMs apparatus will bedistorted, just as in a PIAA coronagraph [Martinache et˜al. (2006)].However the asphericity of the surfaces in the case of ACAD operating onreasonably thin discontinuities, is much smaller than in a PIAAremapping unit. FIG. 21 shows the impact on off-axis PSFs of such adistortion in the worse-case scenario of a geometry similar to JWST. Wedemonstrate that most of the flux remains in the central disk of radiusλ/D for all sources in the field of view of the coronagraphs consideredhere (all the way to 30 cycles per capture). We conclude that, becauseof the small deformations of the DMs, PSF distortion will not be a majorhindrance in exo-planet imaging instruments whose DMs are controlled inorder to mitigate for discontinuities in the aperture.

7.2 Impact of Wavefront Discontinuities in Segmented Telescopes. 7.2.1General Equations in the Presence of Incident Phase Errors andDiscontinuities

So far we have treated primary mirrors' segmentation as a pure amplitudeeffect. In reality the contrast floor in segmented telescopes will bedriven by both phase and amplitude discontinuities: here we explore theimpact of phase errors and discontinuities occurring before two DMswhose surfaces have been set using ACAD. There are two main phenomena tobe considered. The first is the conversion of the incident wavefrontphase before

${DMI}\text{:}\mspace{14mu} \frac{2\; \pi}{\lambda}\Delta \; h_{1}$

into amplitude at the second mirror. The second is the projection ofthis wavefront phase into a remapped phase errors at

${DMI}\text{:}\mspace{14mu} \frac{2\; \pi}{\lambda}\Delta \; {{h_{1}\left( {{f_{1}\left( {x_{1},y_{2}} \right)},{g_{1}\left( {x_{1},y_{2}} \right)}} \right)}.}$

Since the remapping unit is designed using deformable mirrors, both DM1and DM2 a complete correction could be attained in principle. However,the deformable mirrors are continuous while Δh₁ presenteddiscontinuities. Thus, complete corrections for segmented mirrors mightnot be achieved in practice. Below we discuss the following two mainpoints. (1) Even if the phase wavefront error Δh₁ has discontinuities,the phase errors within in segment still drive the phase to amplitudeconversion and thus the propagated amplitude at DM2. In that casetreatments of these phenomenons that have already been discussed in theliterature for monolithic apertures are still valid for small enoughphase errors and smooth enough remapping functions, For ACAD remappingthis smoothness constraint is naturally enforced by the limited numberof actuators across the DM surface. In this case phase to amplitudeconversion can in principle be corrected using DM1. (2) Remapped phasediscontinuities can be corrected for a finite number of spatialfrequencies using a continuous phase sheet deformable mirror. Weillustrate this partial correction over a 20% bandwidth using numericalsimulations of a post-ACAD half dark hole created by superposing a smallperturbation, computed using a linear wavefront control algorithm, tothe ACAD DM2 surface.

If the incoming wavefront is written as Δh₁ and the solution of theMonge-Ampere Equation for DM1 as h₁ ⁰ then one can conduct the analysisin Eq. 9 to 19 using {tilde over (h)}₁=h₁ ⁰+Δh₁. Under the assumptionthat surface of DM2 is set as {tilde over (h)}₂ in order to conserveOptical Path Length then one can re-write the remapping as ({tilde over(f)}₁,{tilde over (g)}₁) defined by:

$\begin{matrix}{\mspace{79mu} {{{\frac{\partial\text{?}}{\partial x}\text{?}} = \frac{{\text{?}\left( {x_{2},y_{2}} \right)} - x_{2}}{Z}}\mspace{79mu} {{\frac{\partial\text{?}}{\partial y}\text{?}} = {{\frac{{\text{?}\left( {x_{2},y_{2}} \right)} - y_{2}}{Z}.\text{?}}\text{indicates text missing or illegible when filed}}}}} & (63)\end{matrix}$

Moreover if edge ringing has been properly mitigated then the ray opticssolution is valid and the field at DM2 can be written as:

$\begin{matrix}{{E_{{DM}\; 2}\left( {x_{2},y_{2}} \right)} = {\left\{ {\left( \frac{E_{{DM}_{1}}}{{\left( {1 + \frac{\partial^{2}\text{?}}{\partial x^{2}}} \right)\left( {1 + \frac{\partial^{2}\text{?}}{\partial y^{2}}} \right)} - \left( \frac{\partial^{2}\text{?}}{\partial y^{2}} \right)^{2}} \right){\text{?}\text{?}}} \right\} _{({x_{2},y_{2}})}{\text{?}\text{indicates text missing or illegible when filed}}}} & (64)\end{matrix}$

7.2.2 Impact on the Amplitude after ACAD

Broadband wavefront correction (20% bandwidth around 700 nm) with asingle DM in segmented telescope with discontinuous surface errors. TopLeft: wavefront before the coronagraph. Top Right: broadband aberratedPSF with DM at rest. Bottom Left: DM surface resulting from thewavefront control algorithm. Bottom Right: broadband corrected PSF. Notethat the wavefront control algorithm seeks to compensate for thediffractive artifacts associated with the secondary support structures:it attenuates them on the right side of the PSF while it strengthensthem on the left side of the PSF. As a result the DM surface becomes toolarge at the pupil spider's location and the quasi-linear wavefrontcontrol algorithm eventually diverges.

We first consider the amplitude profile in Eq. 64. it is composed of twofactors: the remapped telescope aperture, E_(DM1)({tilde over(f)}₁,{tilde over (g)}₁), and the determinant of Id+

[{tilde over (h)}₁].

The first condition necessary for the incoming wavefront not to perturbthe ACAD solution is: Δh₁ is such that the remapping is not modified atthe pupil locations where the telescope aperture is not zero E_(DM1)≠0.This results in the following conditions:

${\frac{{\partial\Delta}\; h_{1}}{\partial x}{\operatorname{<<}\frac{{\partial\Delta}\; h_{1}^{0}}{\partial x}}\mspace{14mu} {for}\mspace{14mu} \left( {x,y} \right)\mspace{14mu} {such}\mspace{14mu} {that}\mspace{14mu} {E_{{DM}\; 1}\left( {x,y} \right)}} \neq 0$${\frac{{\partial\Delta}\; h_{1}}{\partial y}{\operatorname{<<}\frac{{\partial\Delta}\; h_{1}^{0}}{\partial y}}\mspace{14mu} {for}\mspace{14mu} \left( {x,y} \right)\mspace{14mu} {such}\mspace{14mu} {that}\mspace{14mu} {E_{{DM}\; 1}\left( {x,y} \right)}} \neq 0$

At the locations where E_(DM1)=0 there is no light illuminating thediscontinuous wavefront and thus the large local slopes at theselocation have no impact on the remapping functions (f₁,g₁). Theseconditions are not true in segmented telescopes that are not properlyphased, for which the tip-tilt error over each segment can reach severalwaves. However under the assumption that the primary has been properlyphased (for instance the residual rms wavefront after phasing isexpected to be ˜ 1/10th of a wave, similar to values expected for JWSTNIRCAM) then these conditions are true within the boundaries of eachsegment. Moreover, while the local wavefront slopes at the segment'sdiscontinuities do not respect this condition the incident amplitude atthese points is E_(DM1)(x,y)=0 and they thus do not perturb the ACADremapping solution.

The second necessary condition resides in the fact that the determinantof Id+

({tilde over (h)}₁) is not equal to det[Id+

(h₁ ⁰)] at the pupil locations where the telescope aperture is not zeroought not have a severe impact on contrast. One can use thelinearization in [Loeper Rapetti (2005)] to show that:

$\begin{matrix}{\frac{1}{\det \left\lbrack {{Id} + {\mathcal{H}\left( {\overset{\sim}{h}}_{1} \right)}} \right\rbrack} = \frac{1}{{\det \left\lbrack {{Id} + {\mathcal{H}\left( {\overset{\sim}{h}}_{1}^{0} \right)}} \right\rbrack}\left( {1 + \frac{\begin{matrix}{{\left( {1 + h_{1\; {xx}}^{0}} \right)\Delta \; h_{1\; {yy}}} + {\left( {1 + h_{1\; {yy}}^{0}} \right)\Delta \; h_{1\; {xx}}} -} \\{2*h_{1\; {xy}}^{0}\Delta \; h_{1\; {xy}}}\end{matrix}}{\det \left\lbrack {{Id} + {\mathcal{H}\left( h_{1}^{0} \right)}} \right\rbrack}} \right)}} & (65) \\{\mspace{79mu} {\frac{1}{\det \left\lbrack {{Id} + {\mathcal{H}\left( {\overset{\sim}{h}}_{1} \right)}} \right\rbrack} = \frac{1}{{\det \left\lbrack {{Id} + {\mathcal{H}\left( {\overset{\sim}{h}}_{1}^{0} \right)}} \right\rbrack}\left( {1 + {\Delta \; {A\left( {\Delta \; h_{1}} \right)}}} \right.}}} & (66)\end{matrix}$

The perturbation term ΔA(Δh₁) corresponds to the full non-linearexpression of the phase to amplitude conversion of wavefront errors thatoccurs in pupil remapping units. In [Pueyo et˜al. (2011)] we derived asimilar expression in the linear case, when Δh₁<<λ and showed that inthe pupil regions where the the beam is converging this phase toamplitude conversion was enhanced when compared to the case of a Fresnelpropagation. In a recent study [Krist et˜al. (2011)] presentedsimulations predicting that this effect was quite severe in PIAAcoronagraphs and can limit the broadband contrast after wavefrontcontrol unless DMs were placed before the remapping unit. In principleACAD will not suffer from this limitation as the first asphericalsurface of the remapping unit is actually a Deformable Mirror that canactually compensate for Δh₁, before any phase to amplitude wavefrontmodulation occurs. Devising a wavefront controller that relies on DM1requires moreover a computationally efficient model to propagatearbitrary wavefronts through ACAD. Such a tool was developed in [Kristet˜al. (2010)] assuming azimuthally symmetric geometries. Since devisingsuch a tool in ACAD's case, in the asymmetric case, represents asubstantial effort well beyond the problem of prescribing ACAD DMshapes, we chose not to include such simulations in the presentmanuscript. Since we are using Deformable Mirrors with a limited numberof actuators, ACAD remapping is in general less severe than in the caseof PIAA. We thus expect the results regarding the wavefront correctionbefore the remapping unit reported in [Krist et˜al. (2011)] to hold.This is provided that the DM actuators can adequately capture the highspatial frequency content of Δh₁ to create a dark hole in thecoronagraphic PSF. We tackle this particular aspect next when discussingthe case of phase errors, in the absence of wavefront phase to amplitudeconversion. Once again note that while the local wavefront curvaturesare very large at the segment's discontinuities, the incident amplitudeat these points is E_(DM1)(x,y)=0 and thus they do not have an impact onthe ACAD phase to amplitude modulation. In practice if the DM is notexactly located at a location conjugate to the telescope pupil theactual wavefront discontinuities will be slightly illuminated and mightperturb the remapping functions and the phase to amplitude conversion.While this might tighten requirements regarding the positioning of DM1in the direction of the optical axis we do not expect this effect to bea major obstacle to successful ACAD implementations.

7.2.3 Impact on the Phase after ACAD

In practice, when Δh₁ presents discontinuities, the surface of DM2cannot be set to the deformation {tilde over (h)}₂ that conservesOptical Path Length, since we work under the assumption that the DM hasa continuous phase-sheet. While this has no impact on the discussionabove regarding the amplitude of E_(DM2), since DM1 is solelyresponsible for this part, it ought to be taken into account whendiscussing the phase at DM2. Under the assumption that Δh₁ does notperturb the nominal ACAD remapping function then one can show that thephase at DM2 is:

$\begin{matrix}{{\arg \left\lbrack {E_{{DM}\; 2}\left( {x_{2},y_{2}} \right)} \right\rbrack}=={\frac{2\; \pi}{\lambda}\left( {{\Delta \; {h_{1}\left( {{f_{1}^{0}\left( {x_{2},y_{2}} \right)},{g_{1}^{0}\left( {x_{2},y_{2}} \right)}} \right)}} + {\Delta \; {h_{2}\left( {x_{2},y_{2}} \right)}}} \right)}} & (67)\end{matrix}$

where Δh₂(x₂,y₂) is a small continuous surface deformation superposed tothe ACAD shape of DM2 and Δh₁(f₁ ⁰(x₂,y₂), g₁ ⁰(x₂,y₂)) is the telescopeOPD seen through the DM based remapping unit. This second term presentsphase discontinuities whose spatial scale has been contracted by ACAD.When these discontinuities are very small then their high spatialfrequency content does not disrupt the ability of DM2 to correct for lowto mid-spatial frequency wavefront errors wavefront errors. However asthe discontinuities become larger their high spatial frequency contentcan fold into the region of the PSF that the DMs seek to cancel. These“frequency folding” speckles are highly chromatic [Give'on et˜al.(2006)] and can have a severe impact on the spectral bandwidth of acoronagraph whose wavefront is corrected using a continuous DM.

In order to assess the impact of this phenomenon, we conducted a seriesof simulations based on single DM wavefront control algorithm that seeksto create a dark hole in one half of the image plane at in as in [Borda& Traub (2006)]. We use the example of a geometry similar to JWST andwork under the assumption that the discontinuous wavefront incident tothe coronagraph has the same spatial frequency content as a JWST NIRCAMOptical Path Difference that has been adjusted to 70 nm rms in order tomimic a visible Strehl similar to the near-infrared Strehl of JWST. Thenon-linear wavefront and sensing and control problem associated withphasing a primary mirror to such level of precision is undoubtedly acolossal endeavor and is well beyond the scope of this paper. In thissection we work under the assumption that the primary mirror either hasbeen phased to such a level, that the wavefront discontinuities are nolarger than 200 nm peak to valley or that the wavefront has beenotherwise corrected down to this specification using a segmentedDeformable Mirror that is conjugate with the primary mirror. Moreover weassume (1) that the residual post-phasing wavefront map has beencharacterized and can be used in order to build the linear modelunderlying the wavefront controller (2) the focal plane wavefrontestimator (carried using DM diversity as in [Borda. & Traub (2006)] forinstance) is capable to yield an exact estimate of the complexelectrical field at the science camera. Underlying this last assumptionis the overly optimistic premise that wavefront will remain unchangedover the course of each high-contrast exposure. While this is not arealistic assumption one could envision the introduction of specificwavefront sensing schemes, with architectures similar to the onecurrently considered for low order wavefront sensors on monolithicapertures [Guyon et˜al. (2009), Wallace et˜al. (2011)], or using aseparate metrology system. The results presented here are thus limitedto configurations for which segment phasing will be dynamicallycompensated using specific sensing and control beyond the scope of thispaper. As this section merely seeks to address the controllability ofwavefront errors in segmented telescopes we chose to conduct oursimulations with a perfect estimator. Finally we use the strokeminimization wavefront control algorithm presented in [Pueyo et˜al.(2009)] to ensure convergence for as many iteration as possible. Wefirst tested the case of a segmented telescope in the absence of ACAD,using a azimuthally symmetric coronagraph and a single DM. We sough tocreate a Dark Hole between 5 and 28 λ₀/D under a 20% bandwidth withλ₀=700 nm. FIG. 22 shows the results of such a simulation. The DM canindeed correct for the discontinuities over a broadband in one half ofthe image plane. However the wavefront control algorithm seeks tocompensate for the diffractive artifacts associated with the secondarysupport structures: it attenuates them on the right side of the PSFwhile it strengthens them on the left side of the PSF. As a result theDM surface becomes too large at the pupil spider's location and thequasi-linear wavefront control algorithm eventually diverges forcontrasts ˜10⁶.

Broadband wavefront correction (20% bandwidth around 700 nm) in asegmented telescope whose pupil has been re-arranged using ACAD. Thesurface of the first DM is set according to the ACAD equations. Thesurface of the second DM is the sum of the ACAD solution and a smallperturbation calculated using a quasi-linear wavefront controlalgorithm. Top Left: wavefront before the coronagraph. Note that theACAD remapping has compressed the wavefront errors near the struts andthe segment gaps. Top Right: broadband aberrated PSF with DMs set to theACAD solution. Bottom Left: perturbation of DM2's surface resulting fromthe wavefront control algorithm. Bottom Right: broadband corrected PSF.The wavefront control algorithm now yields a DM surface that does notfeature prominent deformations at the location of the spiders. Most ofthe DM stroke is located at the edge of the segments, at location of thewavefront discontinuities. There, the DM surface eventually becomes toolarge and the quasi-linear wavefront control algorithm diverges. Howeverthis occurs higher contrasts than in the absence of ACAD.

We then proceeded to simulate the same configuration in the presence oftwo DMs whose surface at rest was calculated using ACAD. Since theredoes not exist a model yet to propagate arbitrary wavefronts throughACAD (the models in [Krist et˜al. (2011)] only operate under theassumption of an azimuthally symmetric remapping) we can only use thesecond DM for wavefront control. We work under the assumption that theincident wavefront does not perturb the nominal ACAD remapping (which istrue in the case of the surface map we chose for our example) and thatthe arguments in [Krist et˜al. (2010)] hold so that phase to amplitudeconversion in ACAD can be compensated by actuating DM1. Frequencyfolding will then be the phenomenon responsible for the true contrastlimit. In this section we are interested in exploring how this impactsthe controllability of wavefront discontinuities using continuousphase-sheet DMs. We used a azimuthally symmetric coronagraph andsuperposed our wavefront control solution to DM2. We sought to create aDark Hole between 5 and 23 λ₀/D under a 20% bandwidth with λ₀=700 nm.FIG. 22 shows the results of such a simulation. When the incidentwavefront is small enough it is indeed possible to superpose a“classical linear wavefront control” solution to the non-linear ACAD DMshapes in order to carve PSF dark holes. The wavefront control algorithmnow yields a DM surface that does not feature prominent deformations atthe location of the spiders. Most of the DM stroke is located at theedge of the segments, at location of the wavefront discontinuities andseek to correct the frequency folding terms associated with suchdiscontinuities. At these locations the DM surface eventually becomestoo large and the linear wavefront control algorithm diverges. Howeverthis divergence occurs at contrast levels much higher than when the ACADsolution is not applied to the DMs. These simulations show that indeeddiscontinuous phases can be corrected using the second DM of a ACADwhose surfaces have preliminary been set to mitigate the effects ofspiders and segment gaps.

7.3 Ultimate Contrast Limits

Assuming that edge ringing has been properly mitigated so that the rayoptics approximation underlying the calculation of the DMs shapes isvalid, one can wonder about the ultimate contrast limitations of theresults presented in this manuscript. Increasing the number of actuatorswould have dramatic effects on contrast if the actuator count would besuch that N>D/d where d is the scale of the aperture discontinuities.Unfortunately current DM technologies are currently far from such arequirement and the solutions presented here are in the regime whereN<<D/d. In this regime N only has a marginal influence on contrast whencompared to the impact of the cutoff frequency of the tapering kernel.In the regime described here varying the actuator count only changes thesize of the corrected region.

The residual PSF artifacts in FIGS. 11 to 20 follow the direction of theinitial diffraction pattern associated with secondary support structuresand segments. When addressing the problem of aperture discontinuities bysolving the Monge-Ampere Equation, ACAD calculates the DM shapes basedon a pupil plane metric and thus mostly focuses on attenuating thesestructures with little regard to the final contrast. It is actuallyquite remarkable that such a pupil-only approach yields levels ofstarlight extinction of two to three orders of magnitude. A moreappropriate metric would be the final intensity distribution in thepost-coronagraphic image plane. However, as discussed in §. 3 classicalwavefront control algorithms based on a linearization of the DMsdeformations around local equilibrium shapes (such as the ones presentedin [Bora & Traub (2006), Give'on et˜al. (2007)] in the one DM case or[Pueyo et˜al. (2009)] for one or two DMs) cannot be used to compensatethe full aperture discontinuities, This is illustrated in FIG. 22 wherethe DM surface in the vicinity of spiders becomes too large after acertain number of iterations, which leads the iterative algorithm todiverge. When attempting to circumvent this problem by recomputing thelinearization at each iteration, we managed to somewhat stabilize theproblem for a few iterations and reached marginal contrast improvements,but the overall algorithm remained unstable unless a prohibitively smallstep size was used. This is the problem which motivated our effort tocalculate the DM shapes as the full non-linear solution of theMonge-Ampere Equation. While doing so yields significant contrastimprovements in both the case of JWST like geometries, TMT and onaxis-monolithic apertures similar, this approach does not give a properweight to the spatial frequencies of interest for high contrast imaging.We mitigated this effect by giving a strong weight to the spatialfrequencies of interest (in the Dark Hole) when solving the Monge-AmpereEquation.

The next natural step is thus to use non-linear solutions presentedherein to correct for the bulk of the aperture discontinuities and toserve as a starting point for classical linearized waveform controlalgorithms, as illustrated on FIG. 24. FIG. 25 indeed illustrates thatwhen superposing an image plane based wavefront controller to theMonge-Ampere ACAD solution, the contrast can be improved beyond thefloor shown on FIG. 12. However, one DM solutions are of limitedinterest as they only operate efficiently over a finite bandwidth andover half of the image plane. ACAD yields a true broadband solution, andconsequently it would be preferable to use the two DMs in thequasi-linear regime to quantify the true contrast limits of ACAD. Insuch a scheme the DM surfaces are first evaluated as the solution of theMonge-Ampere Equation and then adjusted using the image plane basedwavefront control algorithm presented in [Pueyo et˜al. (2009)]. Howeversuch an exercise requires efficient and robust numerical algorithms toevaluate Eq. 23. Such tool only exist so far in the case of azimuthallysymmetric remapping units [Krist et˜al. (2010)]. Developing suchnumerical tools is thus of primary interest to both quantifying thechromaticity and the true contrast limits achievable with on-axis and/orsegmented telescopes.

8 Conclusion

We have introduced a technique that takes advantage of the presence ofDeformable Mirrors in modern high-contrast coronagraph to compensate foramplitude discontinuities in on-axis and/or segmented telescopes. Ourcalculations predict that this high throughput class of solutionsoperates under broadband illumination even in the presence of reasonablysmall wavefront errors and discontinuities. Our approach relies oncontrolling two sequential Deformable Mirrors in a non-linear regime yetunexplored in the field of high-contrast imaging. Indeed the mirrors'shapes are calculated as the solution of the two-dimensional pupilremapping problem, which can be expressed as a non-linear partialdifferential equation called Monge-Ampere Equation. We called thistechnique Active Compensation of Aperture Discontinuities. While weillustrated the efficiency of ACAD using Apodized Pupil Lyot and PhaseInduced Amplitude Coronagraph, it is applicable to all types ofcoronagraphs and thus enables one to translate the past decade ofinvestigation in coronagraphy with unobscured monolithic apertures to amuch wider class of telescope architectures. Because ACAD consists of asimple remapping of the telescope pupil, it is a true broadbandsolution. Provided that the coronagraph chosen operates under abroadband illumination, ACAD allows high contrast observations over alarge spectral bandwidth as pupil remapping is an achromatic phenomenon.We showed that wavelength edge diffraction artifacts, which are thesource of spectral bandwidth limits in PIAA coronagraphs (also based onpupil remapping), are no larger than classical Fresnel ringing. We thusargued that they will only marginally impact the spectral bandwidth of acoronagraph whose input beam has been corrected with ACAD. The mirrordeformations we find can be achieved, both in curvature and in stroke,with technologies currently used in Ex-AO ground based instruments andin various testbeds aimed at demonstrating high-contrast for space basedapplications. Implementing ACAD on a given on-axis and/or segmented thusdoes not require substantial technology development of criticalcomponents.

For geometries analogous to JWST we have demonstrated that ACAD canachieve at least contrast ˜10⁻⁷, provided that dynamic high precisionsegment phasing can be achieved. For TMT and ELT, ACAD can achieve atleast contrasts ˜10⁻⁸. For on-axis monolithic observatories the designcontrast of the coronagraph can be reached with ACAD when the secondarysupport structures are 5 times thinner than on HST. When they are justas thick as HST contrasts as high as ¹⁰ ⁸ can be reached. These numbersare, however, conservative: an optimal solution can be obtained by finetuning the control term in the Monge-Ampere Equation to thecharacteristic scale of each discontinuity. As our goal was to introducethis technique to the astronomical community and emphasize its broadappeal to a wide class of architectures (JWST,ATLAST,HST,TMT,E-ELT) weleft this observatory specific exercise for future applications.

The true contrast limitation of ACAD resides in the fact that theDeformable Mirrors are controlled using a pre-coronagraph pupil basedmetric. However, as illustrated in FIG. 16, the solution provided byACAD can be used as the starting point for classical linearized waveformcontrol algorithms based in image plane diagnostics. In such a controlstrategy, the surfaces are first evaluated as the solution of theMonge-Ampere Equation and then adjusted using the quasi-linear methodpresented in [Pueyo et˜al. (2009)]. This control strategy requiresefficient and robust numerical algorithms to evaluate the fulldiffractive propagation in the remapped Fresnel regime. All thecontrasts reported here are achieved without aberrations, and we showedthat in practice, quasi-linear DM controls based on images at thescience camera will have to be superposed to the ACAD solutions.Finally, as ACAD is broadly applicable to all types of coronagraphs, theremapped pupil can be used as the entry point to relax the design ofcoronagraphs that do operate on segmented apertures such as discussed in[Carlotti et˜al. (2011), Guyon et˜al. (2010)], also illustrated in FIG.16. ACAD is thus a promising tool for future high contrast imaginginstruments on a wide range of observatories as it will allowastronomers to devise high throughput broadband solutions for a varietyof coronagraphs. It only relies on hardware (Deformable Mirrors) thathave been extensively tested over the past ten years. Finally since ACADcan operate with all type of coronagraphs and it renders the last decadeof research on high-contrast imaging technologies with off-axisunobscured apertures applicable to much broader range of telescopearchitectures.

Acknowledgment of Support and Disclaimer

This work was performed in part under a subcontract with the JetPropulsion Laboratory and funded by NASA through the Sagan FellowshipProgram under Prime Contract No. NAS7-03001. JPL is managed for theNational Aeronautics and Space Administration (NASA) by the CaliforniaInstitute of Technology.

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The embodiments illustrated and discussed in this specification areintended only to teach those skilled in the art how to make and use theinvention. In describing embodiments of the invention, specificterminology is employed for the sake of clarity. However, the inventionis not intended to be limited to the specific terminology so selected.The above-described embodiments of the invention may be modified orvaried, without departing from the invention, as appreciated by thoseskilled in the art in light of the above teachings. It is therefore tobe understood that, within the scope of the claims and theirequivalents, the invention may be practiced otherwise than asspecifically described.

We claim:
 1. An active optical beam shaping system, comprising: a firstdeformable mirror arranged to at least partially intercept an entrancebeam of light and to provide a first reflected beam of light; a seconddeformable mirror arranged to at least partially intercept said firstreflected beam of light from said first deformable mirror and to providea second reflected beam of light; and a signal processing and controlsystem configured to communicate with said first and second deformablemirrors, wherein said signal processing and control system is configuredto provide control signals to said first deformable mirror so as toconfigure a reflecting surface of said first deformable mirror tosubstantially conform to a calculated surface shape, wherein said signalprocessing and control system is further configured to provide controlsignals to said second deformable mirror to configure a reflectingsurface of said second deformable mirror to substantially conform to acalculated surface shape, and wherein said first deformable mirror, saidsecond deformable mirror and said signal processing and control systemtogether provide a large amplitude light modulation range to provide anactively shaped optical beam.
 2. An active optical beam shaping systemaccording to claim 1, wherein said large amplitude modulation range ischaracterized by${\frac{\partial h}{\partial\overset{\rightarrow}{r}}} \geq {O\left( \theta_{s} \right)}$wherein $\frac{\partial h}{\partial\overset{\rightarrow}{r}}$ is afocal gradient of at least one of said first deformable mirror and saidsecond deformable mirror at a point of a respective reflecting surfacethereof, wherein h is a surface deflection of a corresponding one ofsaid first deformable mirror and said second deformable mirror at saidpoint, wherein θ_(s) is a solid angle subtended by a disk of radius δand the distance z between said first deformable mirror and said seconddeformable mirror.
 3. An active optical beam shaping system according toclaim 1, wherein said large amplitude modulation range is greater than1.2% of a fully illuminated portion of said entrance beam of light. 4.An active optical beam shaping system according to claim 3, wherein saidlarge amplitude modulation range is substantially an entire range topermit selectable amplitude modulation of portions of said entrance beamof light anywhere from substantially fully attenuated to substantiallyfully illuminated relative to fully illuminated portions of saidentrance beam of light.
 5. An active optical beam shaping systemaccording to claim 1, further comprising a beam analyzer arranged toreceive at least a portion of said exit beam of light reflected fromsaid second deformable mirror, wherein said beam analyzer is configuredto communicate with said signal processing and control system to providefeedback control of at least one of said first and second deformablemirrors.
 6. An active optical beam shaping system according to claim 1,wherein said signal processing and control system is further configuredto calculate said surface shape of at least one of said first and seconddeformable mirrors based on reverse ray tracing calculations.
 7. Anactive optical beam shaping system according to claim 1, wherein saidsignal processing and control system is further configured to calculatesaid surface shape of at least one of said first and second deformablemirrors based on a solution to Monge-Ampere equations corresponding tosaid active optical beam shaping system.
 8. An optical transmitter,comprising: an optical source; and an optical modulator arranged in anoptical path of said optical source, wherein said optical modulatorcomprises: a first deformable mirror arranged to at least partiallyintercept an entrance beam of light from said optical source and toprovide a first reflected beam of light; a second deformable mirrorarranged to at least partially intercept said first reflected beam oflight from said first deformable mirror and to provide a secondreflected beam of light; and a signal processing and control systemconfigured to communicate with said first and second deformable mirrors,wherein said signal processing and control system is configured toprovide control signals to said first deformable mirror and said seconddeformable mirror to provide at least a portion of said second reflectedbeam of light as an output beam having at least one of a selectableamplitude profile or selectable phase profile.
 9. An optical transmitteraccording to claim 8, wherein said signal processing and control systemis configured to provide control signals to said first deformable mirrorand said second deformable mirror such that said output beam has both aselectable amplitude profile and selectable phase profile.
 10. Anoptical transmitter according to claim 9, wherein said signal processingand control system is configured to provide control signals to saidfirst deformable mirror and said second deformable mirror such that saidoutput beam of light comprises soliton pulses.
 11. An optical receiver,comprising: an optical detector; and an optical filter arranged in anoptical path of said optical detector, wherein said optical filtercomprises: a first deformable mirror arranged to at least partiallyintercept an entrance beam of light being detected to provide a firstreflected beam of light; a second deformable mirror arranged to at leastpartially intercept said first reflected beam of light from said firstdeformable mirror and to provide a second reflected beam of light; and asignal processing and control system configured to communicate with saidfirst and second deformable mirrors, wherein said signal processing andcontrol system is configured to provide control signals to said firstdeformable mirror and said second deformable mirror to provide at leasta portion of said second reflected beam of light as an output beamdirected to said optical detector having at least one of a selectableamplitude profile or selectable phase profile.
 12. An opticalcommunication system, comprising: an optical transmitter; an opticaltransmission path optically coupled to said optical transmitter; and anoptical receiver optically coupled to said optical transmission path,wherein at least one of said optical transmitter, said opticaltransmission path, or said optical receiver comprises an active opticalbeam shaper, and wherein said active optical beam shaper comprises: afirst deformable mirror arranged to at least partially intercept anentrance beam of light to provide a first reflected beam of light; asecond deformable mirror arranged to at least partially intercept saidfirst reflected beam of light from said first deformable mirror and toprovide a second reflected beam of light; and a signal processing andcontrol system configured to communicate with said first and seconddeformable mirrors, wherein said signal processing and control system isconfigured to provide control signals to said first deformable mirrorand said second deformable mirror to provide at least a portion of saidsecond reflected beam of light as an output beam having at least one ofa selectable amplitude profile or selectable phase profile.
 13. Anoptical communication system according to claim 12, wherein said opticaltransmission path is at least one of an atmospheric or substantially avacuum transmission path to provide a free-space optical transmissionpath, wherein said an optical transmitter is arranged to transmit afree-space optical signal, and wherein said optical receiver is arrangedto receive a free-space optical signal such that said opticalcommunication system is a free-space optical communication system. 14.An optical communication system according to claim 12, wherein saidoptical transmission path comprises an optical waveguide arranged toreceive optical signals from said optical receiver and transmit saidoptical signals to said optical receiver.
 15. An optical communicationsystem according to claim 14, wherein said optical waveguide is anoptical fiber such that said optical communication system is afiber-optic communication system.
 16. An optical telescope, comprising:a light collection and magnification system; an active optical beamshaper arranged in an optical path of light collected and magnified bysaid light collection and magnification system; and an optical detectionsystem arranged to receive a beam of light from said active optical beamshaper, wherein said active optical beam shaper comprises: a firstdeformable mirror arranged to at least partially intercept an entrancebeam of light to provide a first reflected beam of light; a seconddeformable mirror arranged to at least partially intercept said firstreflected beam of light from said first deformable mirror and to providea second reflected beam of light; and a signal processing and controlsystem configured to communicate with said first and second deformablemirrors, wherein said signal processing and control system is configuredto provide control signals to said first deformable mirror and saidsecond deformable mirror to provide at least a portion of said secondreflected beam of light as an output beam having at least one of aselectable amplitude profile or selectable phase profile.
 17. A solarfurnace, comprising: a light collection system; an active optical beamshaper arranged in an optical path of light collected by said lightcollection system; and an optical focusing system arranged to receive abeam of light from said active optical beam shaper to be focused on anobject to be heated, wherein said active optical beam shaper comprises:a first deformable mirror arranged to at least partially intercept anentrance beam of light to provide a first reflected beam of light; asecond deformable mirror arranged to at least partially intercept saidfirst reflected beam of light from said first deformable mirror and toprovide a second reflected beam of light; and a signal processing andcontrol system configured to communicate with said first and seconddeformable mirrors, wherein said signal processing and control system isconfigured to provide control signals to said first deformable mirrorand said second deformable mirror to provide at least a portion of saidsecond reflected beam of light as an output beam having at least one ofa selectable amplitude profile or selectable phase profile.
 18. Anoptical pulse-shaping system, comprising: an optical pulse source; afirst diffraction grating disposed in an optical path of said opticalpulse source; an active optical beam shaper disposed in an optical pathof light diffracted from said first diffraction grating; and a seconddiffraction grating disposed in an optical path of light output fromsaid active optical beam shaper, wherein said active optical beam shapercomprises: a first deformable mirror arranged to at least partiallyintercept an entrance beam of light to provide a first reflected beam oflight; a second deformable mirror arranged to at least partiallyintercept said first reflected beam of light from said first deformablemirror and to provide a second reflected beam of light; and a signalprocessing and control system configured to communicate with said firstand second deformable mirrors, wherein said signal processing andcontrol system is configured to provide control signals to said firstdeformable mirror and said second deformable mirror to provide at leasta portion of said second reflected beam of light as an output beamhaving at least one of a selectable amplitude profile or selectablephase profile.